lemon/planarity.h
author Peter Kovacs <kpeter@inf.elte.hu>
Sun, 14 Feb 2010 19:06:07 +0100
changeset 848 e05b2b48515a
parent 797 30cb42e3e43a
child 828 5fd7fafc4470
permissions -rw-r--r--
Improve README and mainpage.dox (#342)
     1 /* -*- mode: C++; indent-tabs-mode: nil; -*-
     2  *
     3  * This file is a part of LEMON, a generic C++ optimization library.
     4  *
     5  * Copyright (C) 2003-2009
     6  * Egervary Jeno Kombinatorikus Optimalizalasi Kutatocsoport
     7  * (Egervary Research Group on Combinatorial Optimization, EGRES).
     8  *
     9  * Permission to use, modify and distribute this software is granted
    10  * provided that this copyright notice appears in all copies. For
    11  * precise terms see the accompanying LICENSE file.
    12  *
    13  * This software is provided "AS IS" with no warranty of any kind,
    14  * express or implied, and with no claim as to its suitability for any
    15  * purpose.
    16  *
    17  */
    18 
    19 #ifndef LEMON_PLANARITY_H
    20 #define LEMON_PLANARITY_H
    21 
    22 /// \ingroup planar
    23 /// \file
    24 /// \brief Planarity checking, embedding, drawing and coloring
    25 
    26 #include <vector>
    27 #include <list>
    28 
    29 #include <lemon/dfs.h>
    30 #include <lemon/bfs.h>
    31 #include <lemon/radix_sort.h>
    32 #include <lemon/maps.h>
    33 #include <lemon/path.h>
    34 #include <lemon/bucket_heap.h>
    35 #include <lemon/adaptors.h>
    36 #include <lemon/edge_set.h>
    37 #include <lemon/color.h>
    38 #include <lemon/dim2.h>
    39 
    40 namespace lemon {
    41 
    42   namespace _planarity_bits {
    43 
    44     template <typename Graph>
    45     struct PlanarityVisitor : DfsVisitor<Graph> {
    46 
    47       TEMPLATE_GRAPH_TYPEDEFS(Graph);
    48 
    49       typedef typename Graph::template NodeMap<Arc> PredMap;
    50 
    51       typedef typename Graph::template EdgeMap<bool> TreeMap;
    52 
    53       typedef typename Graph::template NodeMap<int> OrderMap;
    54       typedef std::vector<Node> OrderList;
    55 
    56       typedef typename Graph::template NodeMap<int> LowMap;
    57       typedef typename Graph::template NodeMap<int> AncestorMap;
    58 
    59       PlanarityVisitor(const Graph& graph,
    60                        PredMap& pred_map, TreeMap& tree_map,
    61                        OrderMap& order_map, OrderList& order_list,
    62                        AncestorMap& ancestor_map, LowMap& low_map)
    63         : _graph(graph), _pred_map(pred_map), _tree_map(tree_map),
    64           _order_map(order_map), _order_list(order_list),
    65           _ancestor_map(ancestor_map), _low_map(low_map) {}
    66 
    67       void reach(const Node& node) {
    68         _order_map[node] = _order_list.size();
    69         _low_map[node] = _order_list.size();
    70         _ancestor_map[node] = _order_list.size();
    71         _order_list.push_back(node);
    72       }
    73 
    74       void discover(const Arc& arc) {
    75         Node source = _graph.source(arc);
    76         Node target = _graph.target(arc);
    77 
    78         _tree_map[arc] = true;
    79         _pred_map[target] = arc;
    80       }
    81 
    82       void examine(const Arc& arc) {
    83         Node source = _graph.source(arc);
    84         Node target = _graph.target(arc);
    85 
    86         if (_order_map[target] < _order_map[source] && !_tree_map[arc]) {
    87           if (_low_map[source] > _order_map[target]) {
    88             _low_map[source] = _order_map[target];
    89           }
    90           if (_ancestor_map[source] > _order_map[target]) {
    91             _ancestor_map[source] = _order_map[target];
    92           }
    93         }
    94       }
    95 
    96       void backtrack(const Arc& arc) {
    97         Node source = _graph.source(arc);
    98         Node target = _graph.target(arc);
    99 
   100         if (_low_map[source] > _low_map[target]) {
   101           _low_map[source] = _low_map[target];
   102         }
   103       }
   104 
   105       const Graph& _graph;
   106       PredMap& _pred_map;
   107       TreeMap& _tree_map;
   108       OrderMap& _order_map;
   109       OrderList& _order_list;
   110       AncestorMap& _ancestor_map;
   111       LowMap& _low_map;
   112     };
   113 
   114     template <typename Graph, bool embedding = true>
   115     struct NodeDataNode {
   116       int prev, next;
   117       int visited;
   118       typename Graph::Arc first;
   119       bool inverted;
   120     };
   121 
   122     template <typename Graph>
   123     struct NodeDataNode<Graph, false> {
   124       int prev, next;
   125       int visited;
   126     };
   127 
   128     template <typename Graph>
   129     struct ChildListNode {
   130       typedef typename Graph::Node Node;
   131       Node first;
   132       Node prev, next;
   133     };
   134 
   135     template <typename Graph>
   136     struct ArcListNode {
   137       typename Graph::Arc prev, next;
   138     };
   139 
   140     template <typename Graph>
   141     class PlanarityChecking {
   142     private:
   143       
   144       TEMPLATE_GRAPH_TYPEDEFS(Graph);
   145 
   146       const Graph& _graph;
   147 
   148     private:
   149       
   150       typedef typename Graph::template NodeMap<Arc> PredMap;
   151       
   152       typedef typename Graph::template EdgeMap<bool> TreeMap;
   153       
   154       typedef typename Graph::template NodeMap<int> OrderMap;
   155       typedef std::vector<Node> OrderList;
   156 
   157       typedef typename Graph::template NodeMap<int> LowMap;
   158       typedef typename Graph::template NodeMap<int> AncestorMap;
   159 
   160       typedef _planarity_bits::NodeDataNode<Graph> NodeDataNode;
   161       typedef std::vector<NodeDataNode> NodeData;
   162 
   163       typedef _planarity_bits::ChildListNode<Graph> ChildListNode;
   164       typedef typename Graph::template NodeMap<ChildListNode> ChildLists;
   165 
   166       typedef typename Graph::template NodeMap<std::list<int> > MergeRoots;
   167 
   168       typedef typename Graph::template NodeMap<bool> EmbedArc;
   169 
   170     public:
   171 
   172       PlanarityChecking(const Graph& graph) : _graph(graph) {}
   173 
   174       bool run() {
   175         typedef _planarity_bits::PlanarityVisitor<Graph> Visitor;
   176 
   177         PredMap pred_map(_graph, INVALID);
   178         TreeMap tree_map(_graph, false);
   179 
   180         OrderMap order_map(_graph, -1);
   181         OrderList order_list;
   182 
   183         AncestorMap ancestor_map(_graph, -1);
   184         LowMap low_map(_graph, -1);
   185 
   186         Visitor visitor(_graph, pred_map, tree_map,
   187                         order_map, order_list, ancestor_map, low_map);
   188         DfsVisit<Graph, Visitor> visit(_graph, visitor);
   189         visit.run();
   190 
   191         ChildLists child_lists(_graph);
   192         createChildLists(tree_map, order_map, low_map, child_lists);
   193 
   194         NodeData node_data(2 * order_list.size());
   195 
   196         EmbedArc embed_arc(_graph, false);
   197 
   198         MergeRoots merge_roots(_graph);
   199 
   200         for (int i = order_list.size() - 1; i >= 0; --i) {
   201 
   202           Node node = order_list[i];
   203 
   204           Node source = node;
   205           for (OutArcIt e(_graph, node); e != INVALID; ++e) {
   206             Node target = _graph.target(e);
   207 
   208             if (order_map[source] < order_map[target] && tree_map[e]) {
   209               initFace(target, node_data, order_map, order_list);
   210             }
   211           }
   212 
   213           for (OutArcIt e(_graph, node); e != INVALID; ++e) {
   214             Node target = _graph.target(e);
   215 
   216             if (order_map[source] < order_map[target] && !tree_map[e]) {
   217               embed_arc[target] = true;
   218               walkUp(target, source, i, pred_map, low_map,
   219                      order_map, order_list, node_data, merge_roots);
   220             }
   221           }
   222 
   223           for (typename MergeRoots::Value::iterator it =
   224                  merge_roots[node].begin(); 
   225                it != merge_roots[node].end(); ++it) {
   226             int rn = *it;
   227             walkDown(rn, i, node_data, order_list, child_lists,
   228                      ancestor_map, low_map, embed_arc, merge_roots);
   229           }
   230           merge_roots[node].clear();
   231 
   232           for (OutArcIt e(_graph, node); e != INVALID; ++e) {
   233             Node target = _graph.target(e);
   234 
   235             if (order_map[source] < order_map[target] && !tree_map[e]) {
   236               if (embed_arc[target]) {
   237                 return false;
   238               }
   239             }
   240           }
   241         }
   242 
   243         return true;
   244       }
   245 
   246     private:
   247 
   248       void createChildLists(const TreeMap& tree_map, const OrderMap& order_map,
   249                             const LowMap& low_map, ChildLists& child_lists) {
   250 
   251         for (NodeIt n(_graph); n != INVALID; ++n) {
   252           Node source = n;
   253 
   254           std::vector<Node> targets;
   255           for (OutArcIt e(_graph, n); e != INVALID; ++e) {
   256             Node target = _graph.target(e);
   257 
   258             if (order_map[source] < order_map[target] && tree_map[e]) {
   259               targets.push_back(target);
   260             }
   261           }
   262 
   263           if (targets.size() == 0) {
   264             child_lists[source].first = INVALID;
   265           } else if (targets.size() == 1) {
   266             child_lists[source].first = targets[0];
   267             child_lists[targets[0]].prev = INVALID;
   268             child_lists[targets[0]].next = INVALID;
   269           } else {
   270             radixSort(targets.begin(), targets.end(), mapToFunctor(low_map));
   271             for (int i = 1; i < int(targets.size()); ++i) {
   272               child_lists[targets[i]].prev = targets[i - 1];
   273               child_lists[targets[i - 1]].next = targets[i];
   274             }
   275             child_lists[targets.back()].next = INVALID;
   276             child_lists[targets.front()].prev = INVALID;
   277             child_lists[source].first = targets.front();
   278           }
   279         }
   280       }
   281 
   282       void walkUp(const Node& node, Node root, int rorder,
   283                   const PredMap& pred_map, const LowMap& low_map,
   284                   const OrderMap& order_map, const OrderList& order_list,
   285                   NodeData& node_data, MergeRoots& merge_roots) {
   286 
   287         int na, nb;
   288         bool da, db;
   289 
   290         na = nb = order_map[node];
   291         da = true; db = false;
   292 
   293         while (true) {
   294 
   295           if (node_data[na].visited == rorder) break;
   296           if (node_data[nb].visited == rorder) break;
   297 
   298           node_data[na].visited = rorder;
   299           node_data[nb].visited = rorder;
   300 
   301           int rn = -1;
   302 
   303           if (na >= int(order_list.size())) {
   304             rn = na;
   305           } else if (nb >= int(order_list.size())) {
   306             rn = nb;
   307           }
   308 
   309           if (rn == -1) {
   310             int nn;
   311 
   312             nn = da ? node_data[na].prev : node_data[na].next;
   313             da = node_data[nn].prev != na;
   314             na = nn;
   315 
   316             nn = db ? node_data[nb].prev : node_data[nb].next;
   317             db = node_data[nn].prev != nb;
   318             nb = nn;
   319 
   320           } else {
   321 
   322             Node rep = order_list[rn - order_list.size()];
   323             Node parent = _graph.source(pred_map[rep]);
   324 
   325             if (low_map[rep] < rorder) {
   326               merge_roots[parent].push_back(rn);
   327             } else {
   328               merge_roots[parent].push_front(rn);
   329             }
   330 
   331             if (parent != root) {
   332               na = nb = order_map[parent];
   333               da = true; db = false;
   334             } else {
   335               break;
   336             }
   337           }
   338         }
   339       }
   340 
   341       void walkDown(int rn, int rorder, NodeData& node_data,
   342                     OrderList& order_list, ChildLists& child_lists,
   343                     AncestorMap& ancestor_map, LowMap& low_map,
   344                     EmbedArc& embed_arc, MergeRoots& merge_roots) {
   345 
   346         std::vector<std::pair<int, bool> > merge_stack;
   347 
   348         for (int di = 0; di < 2; ++di) {
   349           bool rd = di == 0;
   350           int pn = rn;
   351           int n = rd ? node_data[rn].next : node_data[rn].prev;
   352 
   353           while (n != rn) {
   354 
   355             Node node = order_list[n];
   356 
   357             if (embed_arc[node]) {
   358 
   359               // Merging components on the critical path
   360               while (!merge_stack.empty()) {
   361 
   362                 // Component root
   363                 int cn = merge_stack.back().first;
   364                 bool cd = merge_stack.back().second;
   365                 merge_stack.pop_back();
   366 
   367                 // Parent of component
   368                 int dn = merge_stack.back().first;
   369                 bool dd = merge_stack.back().second;
   370                 merge_stack.pop_back();
   371 
   372                 Node parent = order_list[dn];
   373 
   374                 // Erasing from merge_roots
   375                 merge_roots[parent].pop_front();
   376 
   377                 Node child = order_list[cn - order_list.size()];
   378 
   379                 // Erasing from child_lists
   380                 if (child_lists[child].prev != INVALID) {
   381                   child_lists[child_lists[child].prev].next =
   382                     child_lists[child].next;
   383                 } else {
   384                   child_lists[parent].first = child_lists[child].next;
   385                 }
   386 
   387                 if (child_lists[child].next != INVALID) {
   388                   child_lists[child_lists[child].next].prev =
   389                     child_lists[child].prev;
   390                 }
   391 
   392                 // Merging external faces
   393                 {
   394                   int en = cn;
   395                   cn = cd ? node_data[cn].prev : node_data[cn].next;
   396                   cd = node_data[cn].next == en;
   397 
   398                 }
   399 
   400                 if (cd) node_data[cn].next = dn; else node_data[cn].prev = dn;
   401                 if (dd) node_data[dn].prev = cn; else node_data[dn].next = cn;
   402 
   403               }
   404 
   405               bool d = pn == node_data[n].prev;
   406 
   407               if (node_data[n].prev == node_data[n].next &&
   408                   node_data[n].inverted) {
   409                 d = !d;
   410               }
   411 
   412               // Embedding arc into external face
   413               if (rd) node_data[rn].next = n; else node_data[rn].prev = n;
   414               if (d) node_data[n].prev = rn; else node_data[n].next = rn;
   415               pn = rn;
   416 
   417               embed_arc[order_list[n]] = false;
   418             }
   419 
   420             if (!merge_roots[node].empty()) {
   421 
   422               bool d = pn == node_data[n].prev;
   423 
   424               merge_stack.push_back(std::make_pair(n, d));
   425 
   426               int rn = merge_roots[node].front();
   427 
   428               int xn = node_data[rn].next;
   429               Node xnode = order_list[xn];
   430 
   431               int yn = node_data[rn].prev;
   432               Node ynode = order_list[yn];
   433 
   434               bool rd;
   435               if (!external(xnode, rorder, child_lists, 
   436                             ancestor_map, low_map)) {
   437                 rd = true;
   438               } else if (!external(ynode, rorder, child_lists,
   439                                    ancestor_map, low_map)) {
   440                 rd = false;
   441               } else if (pertinent(xnode, embed_arc, merge_roots)) {
   442                 rd = true;
   443               } else {
   444                 rd = false;
   445               }
   446 
   447               merge_stack.push_back(std::make_pair(rn, rd));
   448 
   449               pn = rn;
   450               n = rd ? xn : yn;
   451 
   452             } else if (!external(node, rorder, child_lists,
   453                                  ancestor_map, low_map)) {
   454               int nn = (node_data[n].next != pn ?
   455                         node_data[n].next : node_data[n].prev);
   456 
   457               bool nd = n == node_data[nn].prev;
   458 
   459               if (nd) node_data[nn].prev = pn;
   460               else node_data[nn].next = pn;
   461 
   462               if (n == node_data[pn].prev) node_data[pn].prev = nn;
   463               else node_data[pn].next = nn;
   464 
   465               node_data[nn].inverted =
   466                 (node_data[nn].prev == node_data[nn].next && nd != rd);
   467 
   468               n = nn;
   469             }
   470             else break;
   471 
   472           }
   473 
   474           if (!merge_stack.empty() || n == rn) {
   475             break;
   476           }
   477         }
   478       }
   479 
   480       void initFace(const Node& node, NodeData& node_data,
   481                     const OrderMap& order_map, const OrderList& order_list) {
   482         int n = order_map[node];
   483         int rn = n + order_list.size();
   484 
   485         node_data[n].next = node_data[n].prev = rn;
   486         node_data[rn].next = node_data[rn].prev = n;
   487 
   488         node_data[n].visited = order_list.size();
   489         node_data[rn].visited = order_list.size();
   490 
   491       }
   492 
   493       bool external(const Node& node, int rorder,
   494                     ChildLists& child_lists, AncestorMap& ancestor_map,
   495                     LowMap& low_map) {
   496         Node child = child_lists[node].first;
   497 
   498         if (child != INVALID) {
   499           if (low_map[child] < rorder) return true;
   500         }
   501 
   502         if (ancestor_map[node] < rorder) return true;
   503 
   504         return false;
   505       }
   506 
   507       bool pertinent(const Node& node, const EmbedArc& embed_arc,
   508                      const MergeRoots& merge_roots) {
   509         return !merge_roots[node].empty() || embed_arc[node];
   510       }
   511 
   512     };
   513 
   514   }
   515 
   516   /// \ingroup planar
   517   ///
   518   /// \brief Planarity checking of an undirected simple graph
   519   ///
   520   /// This function implements the Boyer-Myrvold algorithm for
   521   /// planarity checking of an undirected graph. It is a simplified
   522   /// version of the PlanarEmbedding algorithm class because neither
   523   /// the embedding nor the kuratowski subdivisons are not computed.
   524   template <typename GR>
   525   bool checkPlanarity(const GR& graph) {
   526     _planarity_bits::PlanarityChecking<GR> pc(graph);
   527     return pc.run();
   528   }
   529 
   530   /// \ingroup planar
   531   ///
   532   /// \brief Planar embedding of an undirected simple graph
   533   ///
   534   /// This class implements the Boyer-Myrvold algorithm for planar
   535   /// embedding of an undirected graph. The planar embedding is an
   536   /// ordering of the outgoing edges of the nodes, which is a possible
   537   /// configuration to draw the graph in the plane. If there is not
   538   /// such ordering then the graph contains a \f$ K_5 \f$ (full graph
   539   /// with 5 nodes) or a \f$ K_{3,3} \f$ (complete bipartite graph on
   540   /// 3 ANode and 3 BNode) subdivision.
   541   ///
   542   /// The current implementation calculates either an embedding or a
   543   /// Kuratowski subdivision. The running time of the algorithm is 
   544   /// \f$ O(n) \f$.
   545   template <typename Graph>
   546   class PlanarEmbedding {
   547   private:
   548 
   549     TEMPLATE_GRAPH_TYPEDEFS(Graph);
   550 
   551     const Graph& _graph;
   552     typename Graph::template ArcMap<Arc> _embedding;
   553 
   554     typename Graph::template EdgeMap<bool> _kuratowski;
   555 
   556   private:
   557 
   558     typedef typename Graph::template NodeMap<Arc> PredMap;
   559 
   560     typedef typename Graph::template EdgeMap<bool> TreeMap;
   561 
   562     typedef typename Graph::template NodeMap<int> OrderMap;
   563     typedef std::vector<Node> OrderList;
   564 
   565     typedef typename Graph::template NodeMap<int> LowMap;
   566     typedef typename Graph::template NodeMap<int> AncestorMap;
   567 
   568     typedef _planarity_bits::NodeDataNode<Graph> NodeDataNode;
   569     typedef std::vector<NodeDataNode> NodeData;
   570 
   571     typedef _planarity_bits::ChildListNode<Graph> ChildListNode;
   572     typedef typename Graph::template NodeMap<ChildListNode> ChildLists;
   573 
   574     typedef typename Graph::template NodeMap<std::list<int> > MergeRoots;
   575 
   576     typedef typename Graph::template NodeMap<Arc> EmbedArc;
   577 
   578     typedef _planarity_bits::ArcListNode<Graph> ArcListNode;
   579     typedef typename Graph::template ArcMap<ArcListNode> ArcLists;
   580 
   581     typedef typename Graph::template NodeMap<bool> FlipMap;
   582 
   583     typedef typename Graph::template NodeMap<int> TypeMap;
   584 
   585     enum IsolatorNodeType {
   586       HIGHX = 6, LOWX = 7,
   587       HIGHY = 8, LOWY = 9,
   588       ROOT = 10, PERTINENT = 11,
   589       INTERNAL = 12
   590     };
   591 
   592   public:
   593 
   594     /// \brief The map for store of embedding
   595     typedef typename Graph::template ArcMap<Arc> EmbeddingMap;
   596 
   597     /// \brief Constructor
   598     ///
   599     /// \note The graph should be simple, i.e. parallel and loop arc
   600     /// free.
   601     PlanarEmbedding(const Graph& graph)
   602       : _graph(graph), _embedding(_graph), _kuratowski(graph, false) {}
   603 
   604     /// \brief Runs the algorithm.
   605     ///
   606     /// Runs the algorithm.
   607     /// \param kuratowski If the parameter is false, then the
   608     /// algorithm does not compute a Kuratowski subdivision.
   609     ///\return %True when the graph is planar.
   610     bool run(bool kuratowski = true) {
   611       typedef _planarity_bits::PlanarityVisitor<Graph> Visitor;
   612 
   613       PredMap pred_map(_graph, INVALID);
   614       TreeMap tree_map(_graph, false);
   615 
   616       OrderMap order_map(_graph, -1);
   617       OrderList order_list;
   618 
   619       AncestorMap ancestor_map(_graph, -1);
   620       LowMap low_map(_graph, -1);
   621 
   622       Visitor visitor(_graph, pred_map, tree_map,
   623                       order_map, order_list, ancestor_map, low_map);
   624       DfsVisit<Graph, Visitor> visit(_graph, visitor);
   625       visit.run();
   626 
   627       ChildLists child_lists(_graph);
   628       createChildLists(tree_map, order_map, low_map, child_lists);
   629 
   630       NodeData node_data(2 * order_list.size());
   631 
   632       EmbedArc embed_arc(_graph, INVALID);
   633 
   634       MergeRoots merge_roots(_graph);
   635 
   636       ArcLists arc_lists(_graph);
   637 
   638       FlipMap flip_map(_graph, false);
   639 
   640       for (int i = order_list.size() - 1; i >= 0; --i) {
   641 
   642         Node node = order_list[i];
   643 
   644         node_data[i].first = INVALID;
   645 
   646         Node source = node;
   647         for (OutArcIt e(_graph, node); e != INVALID; ++e) {
   648           Node target = _graph.target(e);
   649 
   650           if (order_map[source] < order_map[target] && tree_map[e]) {
   651             initFace(target, arc_lists, node_data,
   652                      pred_map, order_map, order_list);
   653           }
   654         }
   655 
   656         for (OutArcIt e(_graph, node); e != INVALID; ++e) {
   657           Node target = _graph.target(e);
   658 
   659           if (order_map[source] < order_map[target] && !tree_map[e]) {
   660             embed_arc[target] = e;
   661             walkUp(target, source, i, pred_map, low_map,
   662                    order_map, order_list, node_data, merge_roots);
   663           }
   664         }
   665 
   666         for (typename MergeRoots::Value::iterator it =
   667                merge_roots[node].begin(); it != merge_roots[node].end(); ++it) {
   668           int rn = *it;
   669           walkDown(rn, i, node_data, arc_lists, flip_map, order_list,
   670                    child_lists, ancestor_map, low_map, embed_arc, merge_roots);
   671         }
   672         merge_roots[node].clear();
   673 
   674         for (OutArcIt e(_graph, node); e != INVALID; ++e) {
   675           Node target = _graph.target(e);
   676 
   677           if (order_map[source] < order_map[target] && !tree_map[e]) {
   678             if (embed_arc[target] != INVALID) {
   679               if (kuratowski) {
   680                 isolateKuratowski(e, node_data, arc_lists, flip_map,
   681                                   order_map, order_list, pred_map, child_lists,
   682                                   ancestor_map, low_map,
   683                                   embed_arc, merge_roots);
   684               }
   685               return false;
   686             }
   687           }
   688         }
   689       }
   690 
   691       for (int i = 0; i < int(order_list.size()); ++i) {
   692 
   693         mergeRemainingFaces(order_list[i], node_data, order_list, order_map,
   694                             child_lists, arc_lists);
   695         storeEmbedding(order_list[i], node_data, order_map, pred_map,
   696                        arc_lists, flip_map);
   697       }
   698 
   699       return true;
   700     }
   701 
   702     /// \brief Gives back the successor of an arc
   703     ///
   704     /// Gives back the successor of an arc. This function makes
   705     /// possible to query the cyclic order of the outgoing arcs from
   706     /// a node.
   707     Arc next(const Arc& arc) const {
   708       return _embedding[arc];
   709     }
   710 
   711     /// \brief Gives back the calculated embedding map
   712     ///
   713     /// The returned map contains the successor of each arc in the
   714     /// graph.
   715     const EmbeddingMap& embeddingMap() const {
   716       return _embedding;
   717     }
   718 
   719     /// \brief Gives back true if the undirected arc is in the
   720     /// kuratowski subdivision
   721     ///
   722     /// Gives back true if the undirected arc is in the kuratowski
   723     /// subdivision
   724     /// \note The \c run() had to be called with true value.
   725     bool kuratowski(const Edge& edge) {
   726       return _kuratowski[edge];
   727     }
   728 
   729   private:
   730 
   731     void createChildLists(const TreeMap& tree_map, const OrderMap& order_map,
   732                           const LowMap& low_map, ChildLists& child_lists) {
   733 
   734       for (NodeIt n(_graph); n != INVALID; ++n) {
   735         Node source = n;
   736 
   737         std::vector<Node> targets;
   738         for (OutArcIt e(_graph, n); e != INVALID; ++e) {
   739           Node target = _graph.target(e);
   740 
   741           if (order_map[source] < order_map[target] && tree_map[e]) {
   742             targets.push_back(target);
   743           }
   744         }
   745 
   746         if (targets.size() == 0) {
   747           child_lists[source].first = INVALID;
   748         } else if (targets.size() == 1) {
   749           child_lists[source].first = targets[0];
   750           child_lists[targets[0]].prev = INVALID;
   751           child_lists[targets[0]].next = INVALID;
   752         } else {
   753           radixSort(targets.begin(), targets.end(), mapToFunctor(low_map));
   754           for (int i = 1; i < int(targets.size()); ++i) {
   755             child_lists[targets[i]].prev = targets[i - 1];
   756             child_lists[targets[i - 1]].next = targets[i];
   757           }
   758           child_lists[targets.back()].next = INVALID;
   759           child_lists[targets.front()].prev = INVALID;
   760           child_lists[source].first = targets.front();
   761         }
   762       }
   763     }
   764 
   765     void walkUp(const Node& node, Node root, int rorder,
   766                 const PredMap& pred_map, const LowMap& low_map,
   767                 const OrderMap& order_map, const OrderList& order_list,
   768                 NodeData& node_data, MergeRoots& merge_roots) {
   769 
   770       int na, nb;
   771       bool da, db;
   772 
   773       na = nb = order_map[node];
   774       da = true; db = false;
   775 
   776       while (true) {
   777 
   778         if (node_data[na].visited == rorder) break;
   779         if (node_data[nb].visited == rorder) break;
   780 
   781         node_data[na].visited = rorder;
   782         node_data[nb].visited = rorder;
   783 
   784         int rn = -1;
   785 
   786         if (na >= int(order_list.size())) {
   787           rn = na;
   788         } else if (nb >= int(order_list.size())) {
   789           rn = nb;
   790         }
   791 
   792         if (rn == -1) {
   793           int nn;
   794 
   795           nn = da ? node_data[na].prev : node_data[na].next;
   796           da = node_data[nn].prev != na;
   797           na = nn;
   798 
   799           nn = db ? node_data[nb].prev : node_data[nb].next;
   800           db = node_data[nn].prev != nb;
   801           nb = nn;
   802 
   803         } else {
   804 
   805           Node rep = order_list[rn - order_list.size()];
   806           Node parent = _graph.source(pred_map[rep]);
   807 
   808           if (low_map[rep] < rorder) {
   809             merge_roots[parent].push_back(rn);
   810           } else {
   811             merge_roots[parent].push_front(rn);
   812           }
   813 
   814           if (parent != root) {
   815             na = nb = order_map[parent];
   816             da = true; db = false;
   817           } else {
   818             break;
   819           }
   820         }
   821       }
   822     }
   823 
   824     void walkDown(int rn, int rorder, NodeData& node_data,
   825                   ArcLists& arc_lists, FlipMap& flip_map,
   826                   OrderList& order_list, ChildLists& child_lists,
   827                   AncestorMap& ancestor_map, LowMap& low_map,
   828                   EmbedArc& embed_arc, MergeRoots& merge_roots) {
   829 
   830       std::vector<std::pair<int, bool> > merge_stack;
   831 
   832       for (int di = 0; di < 2; ++di) {
   833         bool rd = di == 0;
   834         int pn = rn;
   835         int n = rd ? node_data[rn].next : node_data[rn].prev;
   836 
   837         while (n != rn) {
   838 
   839           Node node = order_list[n];
   840 
   841           if (embed_arc[node] != INVALID) {
   842 
   843             // Merging components on the critical path
   844             while (!merge_stack.empty()) {
   845 
   846               // Component root
   847               int cn = merge_stack.back().first;
   848               bool cd = merge_stack.back().second;
   849               merge_stack.pop_back();
   850 
   851               // Parent of component
   852               int dn = merge_stack.back().first;
   853               bool dd = merge_stack.back().second;
   854               merge_stack.pop_back();
   855 
   856               Node parent = order_list[dn];
   857 
   858               // Erasing from merge_roots
   859               merge_roots[parent].pop_front();
   860 
   861               Node child = order_list[cn - order_list.size()];
   862 
   863               // Erasing from child_lists
   864               if (child_lists[child].prev != INVALID) {
   865                 child_lists[child_lists[child].prev].next =
   866                   child_lists[child].next;
   867               } else {
   868                 child_lists[parent].first = child_lists[child].next;
   869               }
   870 
   871               if (child_lists[child].next != INVALID) {
   872                 child_lists[child_lists[child].next].prev =
   873                   child_lists[child].prev;
   874               }
   875 
   876               // Merging arcs + flipping
   877               Arc de = node_data[dn].first;
   878               Arc ce = node_data[cn].first;
   879 
   880               flip_map[order_list[cn - order_list.size()]] = cd != dd;
   881               if (cd != dd) {
   882                 std::swap(arc_lists[ce].prev, arc_lists[ce].next);
   883                 ce = arc_lists[ce].prev;
   884                 std::swap(arc_lists[ce].prev, arc_lists[ce].next);
   885               }
   886 
   887               {
   888                 Arc dne = arc_lists[de].next;
   889                 Arc cne = arc_lists[ce].next;
   890 
   891                 arc_lists[de].next = cne;
   892                 arc_lists[ce].next = dne;
   893 
   894                 arc_lists[dne].prev = ce;
   895                 arc_lists[cne].prev = de;
   896               }
   897 
   898               if (dd) {
   899                 node_data[dn].first = ce;
   900               }
   901 
   902               // Merging external faces
   903               {
   904                 int en = cn;
   905                 cn = cd ? node_data[cn].prev : node_data[cn].next;
   906                 cd = node_data[cn].next == en;
   907 
   908                  if (node_data[cn].prev == node_data[cn].next &&
   909                     node_data[cn].inverted) {
   910                    cd = !cd;
   911                  }
   912               }
   913 
   914               if (cd) node_data[cn].next = dn; else node_data[cn].prev = dn;
   915               if (dd) node_data[dn].prev = cn; else node_data[dn].next = cn;
   916 
   917             }
   918 
   919             bool d = pn == node_data[n].prev;
   920 
   921             if (node_data[n].prev == node_data[n].next &&
   922                 node_data[n].inverted) {
   923               d = !d;
   924             }
   925 
   926             // Add new arc
   927             {
   928               Arc arc = embed_arc[node];
   929               Arc re = node_data[rn].first;
   930 
   931               arc_lists[arc_lists[re].next].prev = arc;
   932               arc_lists[arc].next = arc_lists[re].next;
   933               arc_lists[arc].prev = re;
   934               arc_lists[re].next = arc;
   935 
   936               if (!rd) {
   937                 node_data[rn].first = arc;
   938               }
   939 
   940               Arc rev = _graph.oppositeArc(arc);
   941               Arc e = node_data[n].first;
   942 
   943               arc_lists[arc_lists[e].next].prev = rev;
   944               arc_lists[rev].next = arc_lists[e].next;
   945               arc_lists[rev].prev = e;
   946               arc_lists[e].next = rev;
   947 
   948               if (d) {
   949                 node_data[n].first = rev;
   950               }
   951 
   952             }
   953 
   954             // Embedding arc into external face
   955             if (rd) node_data[rn].next = n; else node_data[rn].prev = n;
   956             if (d) node_data[n].prev = rn; else node_data[n].next = rn;
   957             pn = rn;
   958 
   959             embed_arc[order_list[n]] = INVALID;
   960           }
   961 
   962           if (!merge_roots[node].empty()) {
   963 
   964             bool d = pn == node_data[n].prev;
   965             if (node_data[n].prev == node_data[n].next &&
   966                 node_data[n].inverted) {
   967               d = !d;
   968             }
   969 
   970             merge_stack.push_back(std::make_pair(n, d));
   971 
   972             int rn = merge_roots[node].front();
   973 
   974             int xn = node_data[rn].next;
   975             Node xnode = order_list[xn];
   976 
   977             int yn = node_data[rn].prev;
   978             Node ynode = order_list[yn];
   979 
   980             bool rd;
   981             if (!external(xnode, rorder, child_lists, ancestor_map, low_map)) {
   982               rd = true;
   983             } else if (!external(ynode, rorder, child_lists,
   984                                  ancestor_map, low_map)) {
   985               rd = false;
   986             } else if (pertinent(xnode, embed_arc, merge_roots)) {
   987               rd = true;
   988             } else {
   989               rd = false;
   990             }
   991 
   992             merge_stack.push_back(std::make_pair(rn, rd));
   993 
   994             pn = rn;
   995             n = rd ? xn : yn;
   996 
   997           } else if (!external(node, rorder, child_lists,
   998                                ancestor_map, low_map)) {
   999             int nn = (node_data[n].next != pn ?
  1000                       node_data[n].next : node_data[n].prev);
  1001 
  1002             bool nd = n == node_data[nn].prev;
  1003 
  1004             if (nd) node_data[nn].prev = pn;
  1005             else node_data[nn].next = pn;
  1006 
  1007             if (n == node_data[pn].prev) node_data[pn].prev = nn;
  1008             else node_data[pn].next = nn;
  1009 
  1010             node_data[nn].inverted =
  1011               (node_data[nn].prev == node_data[nn].next && nd != rd);
  1012 
  1013             n = nn;
  1014           }
  1015           else break;
  1016 
  1017         }
  1018 
  1019         if (!merge_stack.empty() || n == rn) {
  1020           break;
  1021         }
  1022       }
  1023     }
  1024 
  1025     void initFace(const Node& node, ArcLists& arc_lists,
  1026                   NodeData& node_data, const PredMap& pred_map,
  1027                   const OrderMap& order_map, const OrderList& order_list) {
  1028       int n = order_map[node];
  1029       int rn = n + order_list.size();
  1030 
  1031       node_data[n].next = node_data[n].prev = rn;
  1032       node_data[rn].next = node_data[rn].prev = n;
  1033 
  1034       node_data[n].visited = order_list.size();
  1035       node_data[rn].visited = order_list.size();
  1036 
  1037       node_data[n].inverted = false;
  1038       node_data[rn].inverted = false;
  1039 
  1040       Arc arc = pred_map[node];
  1041       Arc rev = _graph.oppositeArc(arc);
  1042 
  1043       node_data[rn].first = arc;
  1044       node_data[n].first = rev;
  1045 
  1046       arc_lists[arc].prev = arc;
  1047       arc_lists[arc].next = arc;
  1048 
  1049       arc_lists[rev].prev = rev;
  1050       arc_lists[rev].next = rev;
  1051 
  1052     }
  1053 
  1054     void mergeRemainingFaces(const Node& node, NodeData& node_data,
  1055                              OrderList& order_list, OrderMap& order_map,
  1056                              ChildLists& child_lists, ArcLists& arc_lists) {
  1057       while (child_lists[node].first != INVALID) {
  1058         int dd = order_map[node];
  1059         Node child = child_lists[node].first;
  1060         int cd = order_map[child] + order_list.size();
  1061         child_lists[node].first = child_lists[child].next;
  1062 
  1063         Arc de = node_data[dd].first;
  1064         Arc ce = node_data[cd].first;
  1065 
  1066         if (de != INVALID) {
  1067           Arc dne = arc_lists[de].next;
  1068           Arc cne = arc_lists[ce].next;
  1069 
  1070           arc_lists[de].next = cne;
  1071           arc_lists[ce].next = dne;
  1072 
  1073           arc_lists[dne].prev = ce;
  1074           arc_lists[cne].prev = de;
  1075         }
  1076 
  1077         node_data[dd].first = ce;
  1078 
  1079       }
  1080     }
  1081 
  1082     void storeEmbedding(const Node& node, NodeData& node_data,
  1083                         OrderMap& order_map, PredMap& pred_map,
  1084                         ArcLists& arc_lists, FlipMap& flip_map) {
  1085 
  1086       if (node_data[order_map[node]].first == INVALID) return;
  1087 
  1088       if (pred_map[node] != INVALID) {
  1089         Node source = _graph.source(pred_map[node]);
  1090         flip_map[node] = flip_map[node] != flip_map[source];
  1091       }
  1092 
  1093       Arc first = node_data[order_map[node]].first;
  1094       Arc prev = first;
  1095 
  1096       Arc arc = flip_map[node] ?
  1097         arc_lists[prev].prev : arc_lists[prev].next;
  1098 
  1099       _embedding[prev] = arc;
  1100 
  1101       while (arc != first) {
  1102         Arc next = arc_lists[arc].prev == prev ?
  1103           arc_lists[arc].next : arc_lists[arc].prev;
  1104         prev = arc; arc = next;
  1105         _embedding[prev] = arc;
  1106       }
  1107     }
  1108 
  1109 
  1110     bool external(const Node& node, int rorder,
  1111                   ChildLists& child_lists, AncestorMap& ancestor_map,
  1112                   LowMap& low_map) {
  1113       Node child = child_lists[node].first;
  1114 
  1115       if (child != INVALID) {
  1116         if (low_map[child] < rorder) return true;
  1117       }
  1118 
  1119       if (ancestor_map[node] < rorder) return true;
  1120 
  1121       return false;
  1122     }
  1123 
  1124     bool pertinent(const Node& node, const EmbedArc& embed_arc,
  1125                    const MergeRoots& merge_roots) {
  1126       return !merge_roots[node].empty() || embed_arc[node] != INVALID;
  1127     }
  1128 
  1129     int lowPoint(const Node& node, OrderMap& order_map, ChildLists& child_lists,
  1130                  AncestorMap& ancestor_map, LowMap& low_map) {
  1131       int low_point;
  1132 
  1133       Node child = child_lists[node].first;
  1134 
  1135       if (child != INVALID) {
  1136         low_point = low_map[child];
  1137       } else {
  1138         low_point = order_map[node];
  1139       }
  1140 
  1141       if (low_point > ancestor_map[node]) {
  1142         low_point = ancestor_map[node];
  1143       }
  1144 
  1145       return low_point;
  1146     }
  1147 
  1148     int findComponentRoot(Node root, Node node, ChildLists& child_lists,
  1149                           OrderMap& order_map, OrderList& order_list) {
  1150 
  1151       int order = order_map[root];
  1152       int norder = order_map[node];
  1153 
  1154       Node child = child_lists[root].first;
  1155       while (child != INVALID) {
  1156         int corder = order_map[child];
  1157         if (corder > order && corder < norder) {
  1158           order = corder;
  1159         }
  1160         child = child_lists[child].next;
  1161       }
  1162       return order + order_list.size();
  1163     }
  1164 
  1165     Node findPertinent(Node node, OrderMap& order_map, NodeData& node_data,
  1166                        EmbedArc& embed_arc, MergeRoots& merge_roots) {
  1167       Node wnode =_graph.target(node_data[order_map[node]].first);
  1168       while (!pertinent(wnode, embed_arc, merge_roots)) {
  1169         wnode = _graph.target(node_data[order_map[wnode]].first);
  1170       }
  1171       return wnode;
  1172     }
  1173 
  1174 
  1175     Node findExternal(Node node, int rorder, OrderMap& order_map,
  1176                       ChildLists& child_lists, AncestorMap& ancestor_map,
  1177                       LowMap& low_map, NodeData& node_data) {
  1178       Node wnode =_graph.target(node_data[order_map[node]].first);
  1179       while (!external(wnode, rorder, child_lists, ancestor_map, low_map)) {
  1180         wnode = _graph.target(node_data[order_map[wnode]].first);
  1181       }
  1182       return wnode;
  1183     }
  1184 
  1185     void markCommonPath(Node node, int rorder, Node& wnode, Node& znode,
  1186                         OrderList& order_list, OrderMap& order_map,
  1187                         NodeData& node_data, ArcLists& arc_lists,
  1188                         EmbedArc& embed_arc, MergeRoots& merge_roots,
  1189                         ChildLists& child_lists, AncestorMap& ancestor_map,
  1190                         LowMap& low_map) {
  1191 
  1192       Node cnode = node;
  1193       Node pred = INVALID;
  1194 
  1195       while (true) {
  1196 
  1197         bool pert = pertinent(cnode, embed_arc, merge_roots);
  1198         bool ext = external(cnode, rorder, child_lists, ancestor_map, low_map);
  1199 
  1200         if (pert && ext) {
  1201           if (!merge_roots[cnode].empty()) {
  1202             int cn = merge_roots[cnode].back();
  1203 
  1204             if (low_map[order_list[cn - order_list.size()]] < rorder) {
  1205               Arc arc = node_data[cn].first;
  1206               _kuratowski.set(arc, true);
  1207 
  1208               pred = cnode;
  1209               cnode = _graph.target(arc);
  1210 
  1211               continue;
  1212             }
  1213           }
  1214           wnode = znode = cnode;
  1215           return;
  1216 
  1217         } else if (pert) {
  1218           wnode = cnode;
  1219 
  1220           while (!external(cnode, rorder, child_lists, ancestor_map, low_map)) {
  1221             Arc arc = node_data[order_map[cnode]].first;
  1222 
  1223             if (_graph.target(arc) == pred) {
  1224               arc = arc_lists[arc].next;
  1225             }
  1226             _kuratowski.set(arc, true);
  1227 
  1228             Node next = _graph.target(arc);
  1229             pred = cnode; cnode = next;
  1230           }
  1231 
  1232           znode = cnode;
  1233           return;
  1234 
  1235         } else if (ext) {
  1236           znode = cnode;
  1237 
  1238           while (!pertinent(cnode, embed_arc, merge_roots)) {
  1239             Arc arc = node_data[order_map[cnode]].first;
  1240 
  1241             if (_graph.target(arc) == pred) {
  1242               arc = arc_lists[arc].next;
  1243             }
  1244             _kuratowski.set(arc, true);
  1245 
  1246             Node next = _graph.target(arc);
  1247             pred = cnode; cnode = next;
  1248           }
  1249 
  1250           wnode = cnode;
  1251           return;
  1252 
  1253         } else {
  1254           Arc arc = node_data[order_map[cnode]].first;
  1255 
  1256           if (_graph.target(arc) == pred) {
  1257             arc = arc_lists[arc].next;
  1258           }
  1259           _kuratowski.set(arc, true);
  1260 
  1261           Node next = _graph.target(arc);
  1262           pred = cnode; cnode = next;
  1263         }
  1264 
  1265       }
  1266 
  1267     }
  1268 
  1269     void orientComponent(Node root, int rn, OrderMap& order_map,
  1270                          PredMap& pred_map, NodeData& node_data,
  1271                          ArcLists& arc_lists, FlipMap& flip_map,
  1272                          TypeMap& type_map) {
  1273       node_data[order_map[root]].first = node_data[rn].first;
  1274       type_map[root] = 1;
  1275 
  1276       std::vector<Node> st, qu;
  1277 
  1278       st.push_back(root);
  1279       while (!st.empty()) {
  1280         Node node = st.back();
  1281         st.pop_back();
  1282         qu.push_back(node);
  1283 
  1284         Arc arc = node_data[order_map[node]].first;
  1285 
  1286         if (type_map[_graph.target(arc)] == 0) {
  1287           st.push_back(_graph.target(arc));
  1288           type_map[_graph.target(arc)] = 1;
  1289         }
  1290 
  1291         Arc last = arc, pred = arc;
  1292         arc = arc_lists[arc].next;
  1293         while (arc != last) {
  1294 
  1295           if (type_map[_graph.target(arc)] == 0) {
  1296             st.push_back(_graph.target(arc));
  1297             type_map[_graph.target(arc)] = 1;
  1298           }
  1299 
  1300           Arc next = arc_lists[arc].next != pred ?
  1301             arc_lists[arc].next : arc_lists[arc].prev;
  1302           pred = arc; arc = next;
  1303         }
  1304 
  1305       }
  1306 
  1307       type_map[root] = 2;
  1308       flip_map[root] = false;
  1309 
  1310       for (int i = 1; i < int(qu.size()); ++i) {
  1311 
  1312         Node node = qu[i];
  1313 
  1314         while (type_map[node] != 2) {
  1315           st.push_back(node);
  1316           type_map[node] = 2;
  1317           node = _graph.source(pred_map[node]);
  1318         }
  1319 
  1320         bool flip = flip_map[node];
  1321 
  1322         while (!st.empty()) {
  1323           node = st.back();
  1324           st.pop_back();
  1325 
  1326           flip_map[node] = flip != flip_map[node];
  1327           flip = flip_map[node];
  1328 
  1329           if (flip) {
  1330             Arc arc = node_data[order_map[node]].first;
  1331             std::swap(arc_lists[arc].prev, arc_lists[arc].next);
  1332             arc = arc_lists[arc].prev;
  1333             std::swap(arc_lists[arc].prev, arc_lists[arc].next);
  1334             node_data[order_map[node]].first = arc;
  1335           }
  1336         }
  1337       }
  1338 
  1339       for (int i = 0; i < int(qu.size()); ++i) {
  1340 
  1341         Arc arc = node_data[order_map[qu[i]]].first;
  1342         Arc last = arc, pred = arc;
  1343 
  1344         arc = arc_lists[arc].next;
  1345         while (arc != last) {
  1346 
  1347           if (arc_lists[arc].next == pred) {
  1348             std::swap(arc_lists[arc].next, arc_lists[arc].prev);
  1349           }
  1350           pred = arc; arc = arc_lists[arc].next;
  1351         }
  1352 
  1353       }
  1354     }
  1355 
  1356     void setFaceFlags(Node root, Node wnode, Node ynode, Node xnode,
  1357                       OrderMap& order_map, NodeData& node_data,
  1358                       TypeMap& type_map) {
  1359       Node node = _graph.target(node_data[order_map[root]].first);
  1360 
  1361       while (node != ynode) {
  1362         type_map[node] = HIGHY;
  1363         node = _graph.target(node_data[order_map[node]].first);
  1364       }
  1365 
  1366       while (node != wnode) {
  1367         type_map[node] = LOWY;
  1368         node = _graph.target(node_data[order_map[node]].first);
  1369       }
  1370 
  1371       node = _graph.target(node_data[order_map[wnode]].first);
  1372 
  1373       while (node != xnode) {
  1374         type_map[node] = LOWX;
  1375         node = _graph.target(node_data[order_map[node]].first);
  1376       }
  1377       type_map[node] = LOWX;
  1378 
  1379       node = _graph.target(node_data[order_map[xnode]].first);
  1380       while (node != root) {
  1381         type_map[node] = HIGHX;
  1382         node = _graph.target(node_data[order_map[node]].first);
  1383       }
  1384 
  1385       type_map[wnode] = PERTINENT;
  1386       type_map[root] = ROOT;
  1387     }
  1388 
  1389     void findInternalPath(std::vector<Arc>& ipath,
  1390                           Node wnode, Node root, TypeMap& type_map,
  1391                           OrderMap& order_map, NodeData& node_data,
  1392                           ArcLists& arc_lists) {
  1393       std::vector<Arc> st;
  1394 
  1395       Node node = wnode;
  1396 
  1397       while (node != root) {
  1398         Arc arc = arc_lists[node_data[order_map[node]].first].next;
  1399         st.push_back(arc);
  1400         node = _graph.target(arc);
  1401       }
  1402 
  1403       while (true) {
  1404         Arc arc = st.back();
  1405         if (type_map[_graph.target(arc)] == LOWX ||
  1406             type_map[_graph.target(arc)] == HIGHX) {
  1407           break;
  1408         }
  1409         if (type_map[_graph.target(arc)] == 2) {
  1410           type_map[_graph.target(arc)] = 3;
  1411 
  1412           arc = arc_lists[_graph.oppositeArc(arc)].next;
  1413           st.push_back(arc);
  1414         } else {
  1415           st.pop_back();
  1416           arc = arc_lists[arc].next;
  1417 
  1418           while (_graph.oppositeArc(arc) == st.back()) {
  1419             arc = st.back();
  1420             st.pop_back();
  1421             arc = arc_lists[arc].next;
  1422           }
  1423           st.push_back(arc);
  1424         }
  1425       }
  1426 
  1427       for (int i = 0; i < int(st.size()); ++i) {
  1428         if (type_map[_graph.target(st[i])] != LOWY &&
  1429             type_map[_graph.target(st[i])] != HIGHY) {
  1430           for (; i < int(st.size()); ++i) {
  1431             ipath.push_back(st[i]);
  1432           }
  1433         }
  1434       }
  1435     }
  1436 
  1437     void setInternalFlags(std::vector<Arc>& ipath, TypeMap& type_map) {
  1438       for (int i = 1; i < int(ipath.size()); ++i) {
  1439         type_map[_graph.source(ipath[i])] = INTERNAL;
  1440       }
  1441     }
  1442 
  1443     void findPilePath(std::vector<Arc>& ppath,
  1444                       Node root, TypeMap& type_map, OrderMap& order_map,
  1445                       NodeData& node_data, ArcLists& arc_lists) {
  1446       std::vector<Arc> st;
  1447 
  1448       st.push_back(_graph.oppositeArc(node_data[order_map[root]].first));
  1449       st.push_back(node_data[order_map[root]].first);
  1450 
  1451       while (st.size() > 1) {
  1452         Arc arc = st.back();
  1453         if (type_map[_graph.target(arc)] == INTERNAL) {
  1454           break;
  1455         }
  1456         if (type_map[_graph.target(arc)] == 3) {
  1457           type_map[_graph.target(arc)] = 4;
  1458 
  1459           arc = arc_lists[_graph.oppositeArc(arc)].next;
  1460           st.push_back(arc);
  1461         } else {
  1462           st.pop_back();
  1463           arc = arc_lists[arc].next;
  1464 
  1465           while (!st.empty() && _graph.oppositeArc(arc) == st.back()) {
  1466             arc = st.back();
  1467             st.pop_back();
  1468             arc = arc_lists[arc].next;
  1469           }
  1470           st.push_back(arc);
  1471         }
  1472       }
  1473 
  1474       for (int i = 1; i < int(st.size()); ++i) {
  1475         ppath.push_back(st[i]);
  1476       }
  1477     }
  1478 
  1479 
  1480     int markExternalPath(Node node, OrderMap& order_map,
  1481                          ChildLists& child_lists, PredMap& pred_map,
  1482                          AncestorMap& ancestor_map, LowMap& low_map) {
  1483       int lp = lowPoint(node, order_map, child_lists,
  1484                         ancestor_map, low_map);
  1485 
  1486       if (ancestor_map[node] != lp) {
  1487         node = child_lists[node].first;
  1488         _kuratowski[pred_map[node]] = true;
  1489 
  1490         while (ancestor_map[node] != lp) {
  1491           for (OutArcIt e(_graph, node); e != INVALID; ++e) {
  1492             Node tnode = _graph.target(e);
  1493             if (order_map[tnode] > order_map[node] && low_map[tnode] == lp) {
  1494               node = tnode;
  1495               _kuratowski[e] = true;
  1496               break;
  1497             }
  1498           }
  1499         }
  1500       }
  1501 
  1502       for (OutArcIt e(_graph, node); e != INVALID; ++e) {
  1503         if (order_map[_graph.target(e)] == lp) {
  1504           _kuratowski[e] = true;
  1505           break;
  1506         }
  1507       }
  1508 
  1509       return lp;
  1510     }
  1511 
  1512     void markPertinentPath(Node node, OrderMap& order_map,
  1513                            NodeData& node_data, ArcLists& arc_lists,
  1514                            EmbedArc& embed_arc, MergeRoots& merge_roots) {
  1515       while (embed_arc[node] == INVALID) {
  1516         int n = merge_roots[node].front();
  1517         Arc arc = node_data[n].first;
  1518 
  1519         _kuratowski.set(arc, true);
  1520 
  1521         Node pred = node;
  1522         node = _graph.target(arc);
  1523         while (!pertinent(node, embed_arc, merge_roots)) {
  1524           arc = node_data[order_map[node]].first;
  1525           if (_graph.target(arc) == pred) {
  1526             arc = arc_lists[arc].next;
  1527           }
  1528           _kuratowski.set(arc, true);
  1529           pred = node;
  1530           node = _graph.target(arc);
  1531         }
  1532       }
  1533       _kuratowski.set(embed_arc[node], true);
  1534     }
  1535 
  1536     void markPredPath(Node node, Node snode, PredMap& pred_map) {
  1537       while (node != snode) {
  1538         _kuratowski.set(pred_map[node], true);
  1539         node = _graph.source(pred_map[node]);
  1540       }
  1541     }
  1542 
  1543     void markFacePath(Node ynode, Node xnode,
  1544                       OrderMap& order_map, NodeData& node_data) {
  1545       Arc arc = node_data[order_map[ynode]].first;
  1546       Node node = _graph.target(arc);
  1547       _kuratowski.set(arc, true);
  1548 
  1549       while (node != xnode) {
  1550         arc = node_data[order_map[node]].first;
  1551         _kuratowski.set(arc, true);
  1552         node = _graph.target(arc);
  1553       }
  1554     }
  1555 
  1556     void markInternalPath(std::vector<Arc>& path) {
  1557       for (int i = 0; i < int(path.size()); ++i) {
  1558         _kuratowski.set(path[i], true);
  1559       }
  1560     }
  1561 
  1562     void markPilePath(std::vector<Arc>& path) {
  1563       for (int i = 0; i < int(path.size()); ++i) {
  1564         _kuratowski.set(path[i], true);
  1565       }
  1566     }
  1567 
  1568     void isolateKuratowski(Arc arc, NodeData& node_data,
  1569                            ArcLists& arc_lists, FlipMap& flip_map,
  1570                            OrderMap& order_map, OrderList& order_list,
  1571                            PredMap& pred_map, ChildLists& child_lists,
  1572                            AncestorMap& ancestor_map, LowMap& low_map,
  1573                            EmbedArc& embed_arc, MergeRoots& merge_roots) {
  1574 
  1575       Node root = _graph.source(arc);
  1576       Node enode = _graph.target(arc);
  1577 
  1578       int rorder = order_map[root];
  1579 
  1580       TypeMap type_map(_graph, 0);
  1581 
  1582       int rn = findComponentRoot(root, enode, child_lists,
  1583                                  order_map, order_list);
  1584 
  1585       Node xnode = order_list[node_data[rn].next];
  1586       Node ynode = order_list[node_data[rn].prev];
  1587 
  1588       // Minor-A
  1589       {
  1590         while (!merge_roots[xnode].empty() || !merge_roots[ynode].empty()) {
  1591 
  1592           if (!merge_roots[xnode].empty()) {
  1593             root = xnode;
  1594             rn = merge_roots[xnode].front();
  1595           } else {
  1596             root = ynode;
  1597             rn = merge_roots[ynode].front();
  1598           }
  1599 
  1600           xnode = order_list[node_data[rn].next];
  1601           ynode = order_list[node_data[rn].prev];
  1602         }
  1603 
  1604         if (root != _graph.source(arc)) {
  1605           orientComponent(root, rn, order_map, pred_map,
  1606                           node_data, arc_lists, flip_map, type_map);
  1607           markFacePath(root, root, order_map, node_data);
  1608           int xlp = markExternalPath(xnode, order_map, child_lists,
  1609                                      pred_map, ancestor_map, low_map);
  1610           int ylp = markExternalPath(ynode, order_map, child_lists,
  1611                                      pred_map, ancestor_map, low_map);
  1612           markPredPath(root, order_list[xlp < ylp ? xlp : ylp], pred_map);
  1613           Node lwnode = findPertinent(ynode, order_map, node_data,
  1614                                       embed_arc, merge_roots);
  1615 
  1616           markPertinentPath(lwnode, order_map, node_data, arc_lists,
  1617                             embed_arc, merge_roots);
  1618 
  1619           return;
  1620         }
  1621       }
  1622 
  1623       orientComponent(root, rn, order_map, pred_map,
  1624                       node_data, arc_lists, flip_map, type_map);
  1625 
  1626       Node wnode = findPertinent(ynode, order_map, node_data,
  1627                                  embed_arc, merge_roots);
  1628       setFaceFlags(root, wnode, ynode, xnode, order_map, node_data, type_map);
  1629 
  1630 
  1631       //Minor-B
  1632       if (!merge_roots[wnode].empty()) {
  1633         int cn = merge_roots[wnode].back();
  1634         Node rep = order_list[cn - order_list.size()];
  1635         if (low_map[rep] < rorder) {
  1636           markFacePath(root, root, order_map, node_data);
  1637           int xlp = markExternalPath(xnode, order_map, child_lists,
  1638                                      pred_map, ancestor_map, low_map);
  1639           int ylp = markExternalPath(ynode, order_map, child_lists,
  1640                                      pred_map, ancestor_map, low_map);
  1641 
  1642           Node lwnode, lznode;
  1643           markCommonPath(wnode, rorder, lwnode, lznode, order_list,
  1644                          order_map, node_data, arc_lists, embed_arc,
  1645                          merge_roots, child_lists, ancestor_map, low_map);
  1646 
  1647           markPertinentPath(lwnode, order_map, node_data, arc_lists,
  1648                             embed_arc, merge_roots);
  1649           int zlp = markExternalPath(lznode, order_map, child_lists,
  1650                                      pred_map, ancestor_map, low_map);
  1651 
  1652           int minlp = xlp < ylp ? xlp : ylp;
  1653           if (zlp < minlp) minlp = zlp;
  1654 
  1655           int maxlp = xlp > ylp ? xlp : ylp;
  1656           if (zlp > maxlp) maxlp = zlp;
  1657 
  1658           markPredPath(order_list[maxlp], order_list[minlp], pred_map);
  1659 
  1660           return;
  1661         }
  1662       }
  1663 
  1664       Node pxnode, pynode;
  1665       std::vector<Arc> ipath;
  1666       findInternalPath(ipath, wnode, root, type_map, order_map,
  1667                        node_data, arc_lists);
  1668       setInternalFlags(ipath, type_map);
  1669       pynode = _graph.source(ipath.front());
  1670       pxnode = _graph.target(ipath.back());
  1671 
  1672       wnode = findPertinent(pynode, order_map, node_data,
  1673                             embed_arc, merge_roots);
  1674 
  1675       // Minor-C
  1676       {
  1677         if (type_map[_graph.source(ipath.front())] == HIGHY) {
  1678           if (type_map[_graph.target(ipath.back())] == HIGHX) {
  1679             markFacePath(xnode, pxnode, order_map, node_data);
  1680           }
  1681           markFacePath(root, xnode, order_map, node_data);
  1682           markPertinentPath(wnode, order_map, node_data, arc_lists,
  1683                             embed_arc, merge_roots);
  1684           markInternalPath(ipath);
  1685           int xlp = markExternalPath(xnode, order_map, child_lists,
  1686                                      pred_map, ancestor_map, low_map);
  1687           int ylp = markExternalPath(ynode, order_map, child_lists,
  1688                                      pred_map, ancestor_map, low_map);
  1689           markPredPath(root, order_list[xlp < ylp ? xlp : ylp], pred_map);
  1690           return;
  1691         }
  1692 
  1693         if (type_map[_graph.target(ipath.back())] == HIGHX) {
  1694           markFacePath(ynode, root, order_map, node_data);
  1695           markPertinentPath(wnode, order_map, node_data, arc_lists,
  1696                             embed_arc, merge_roots);
  1697           markInternalPath(ipath);
  1698           int xlp = markExternalPath(xnode, order_map, child_lists,
  1699                                      pred_map, ancestor_map, low_map);
  1700           int ylp = markExternalPath(ynode, order_map, child_lists,
  1701                                      pred_map, ancestor_map, low_map);
  1702           markPredPath(root, order_list[xlp < ylp ? xlp : ylp], pred_map);
  1703           return;
  1704         }
  1705       }
  1706 
  1707       std::vector<Arc> ppath;
  1708       findPilePath(ppath, root, type_map, order_map, node_data, arc_lists);
  1709 
  1710       // Minor-D
  1711       if (!ppath.empty()) {
  1712         markFacePath(ynode, xnode, order_map, node_data);
  1713         markPertinentPath(wnode, order_map, node_data, arc_lists,
  1714                           embed_arc, merge_roots);
  1715         markPilePath(ppath);
  1716         markInternalPath(ipath);
  1717         int xlp = markExternalPath(xnode, order_map, child_lists,
  1718                                    pred_map, ancestor_map, low_map);
  1719         int ylp = markExternalPath(ynode, order_map, child_lists,
  1720                                    pred_map, ancestor_map, low_map);
  1721         markPredPath(root, order_list[xlp < ylp ? xlp : ylp], pred_map);
  1722         return;
  1723       }
  1724 
  1725       // Minor-E*
  1726       {
  1727 
  1728         if (!external(wnode, rorder, child_lists, ancestor_map, low_map)) {
  1729           Node znode = findExternal(pynode, rorder, order_map,
  1730                                     child_lists, ancestor_map,
  1731                                     low_map, node_data);
  1732 
  1733           if (type_map[znode] == LOWY) {
  1734             markFacePath(root, xnode, order_map, node_data);
  1735             markPertinentPath(wnode, order_map, node_data, arc_lists,
  1736                               embed_arc, merge_roots);
  1737             markInternalPath(ipath);
  1738             int xlp = markExternalPath(xnode, order_map, child_lists,
  1739                                        pred_map, ancestor_map, low_map);
  1740             int zlp = markExternalPath(znode, order_map, child_lists,
  1741                                        pred_map, ancestor_map, low_map);
  1742             markPredPath(root, order_list[xlp < zlp ? xlp : zlp], pred_map);
  1743           } else {
  1744             markFacePath(ynode, root, order_map, node_data);
  1745             markPertinentPath(wnode, order_map, node_data, arc_lists,
  1746                               embed_arc, merge_roots);
  1747             markInternalPath(ipath);
  1748             int ylp = markExternalPath(ynode, order_map, child_lists,
  1749                                        pred_map, ancestor_map, low_map);
  1750             int zlp = markExternalPath(znode, order_map, child_lists,
  1751                                        pred_map, ancestor_map, low_map);
  1752             markPredPath(root, order_list[ylp < zlp ? ylp : zlp], pred_map);
  1753           }
  1754           return;
  1755         }
  1756 
  1757         int xlp = markExternalPath(xnode, order_map, child_lists,
  1758                                    pred_map, ancestor_map, low_map);
  1759         int ylp = markExternalPath(ynode, order_map, child_lists,
  1760                                    pred_map, ancestor_map, low_map);
  1761         int wlp = markExternalPath(wnode, order_map, child_lists,
  1762                                    pred_map, ancestor_map, low_map);
  1763 
  1764         if (wlp > xlp && wlp > ylp) {
  1765           markFacePath(root, root, order_map, node_data);
  1766           markPredPath(root, order_list[xlp < ylp ? xlp : ylp], pred_map);
  1767           return;
  1768         }
  1769 
  1770         markInternalPath(ipath);
  1771         markPertinentPath(wnode, order_map, node_data, arc_lists,
  1772                           embed_arc, merge_roots);
  1773 
  1774         if (xlp > ylp && xlp > wlp) {
  1775           markFacePath(root, pynode, order_map, node_data);
  1776           markFacePath(wnode, xnode, order_map, node_data);
  1777           markPredPath(root, order_list[ylp < wlp ? ylp : wlp], pred_map);
  1778           return;
  1779         }
  1780 
  1781         if (ylp > xlp && ylp > wlp) {
  1782           markFacePath(pxnode, root, order_map, node_data);
  1783           markFacePath(ynode, wnode, order_map, node_data);
  1784           markPredPath(root, order_list[xlp < wlp ? xlp : wlp], pred_map);
  1785           return;
  1786         }
  1787 
  1788         if (pynode != ynode) {
  1789           markFacePath(pxnode, wnode, order_map, node_data);
  1790 
  1791           int minlp = xlp < ylp ? xlp : ylp;
  1792           if (wlp < minlp) minlp = wlp;
  1793 
  1794           int maxlp = xlp > ylp ? xlp : ylp;
  1795           if (wlp > maxlp) maxlp = wlp;
  1796 
  1797           markPredPath(order_list[maxlp], order_list[minlp], pred_map);
  1798           return;
  1799         }
  1800 
  1801         if (pxnode != xnode) {
  1802           markFacePath(wnode, pynode, order_map, node_data);
  1803 
  1804           int minlp = xlp < ylp ? xlp : ylp;
  1805           if (wlp < minlp) minlp = wlp;
  1806 
  1807           int maxlp = xlp > ylp ? xlp : ylp;
  1808           if (wlp > maxlp) maxlp = wlp;
  1809 
  1810           markPredPath(order_list[maxlp], order_list[minlp], pred_map);
  1811           return;
  1812         }
  1813 
  1814         markFacePath(root, root, order_map, node_data);
  1815         int minlp = xlp < ylp ? xlp : ylp;
  1816         if (wlp < minlp) minlp = wlp;
  1817         markPredPath(root, order_list[minlp], pred_map);
  1818         return;
  1819       }
  1820 
  1821     }
  1822 
  1823   };
  1824 
  1825   namespace _planarity_bits {
  1826 
  1827     template <typename Graph, typename EmbeddingMap>
  1828     void makeConnected(Graph& graph, EmbeddingMap& embedding) {
  1829       DfsVisitor<Graph> null_visitor;
  1830       DfsVisit<Graph, DfsVisitor<Graph> > dfs(graph, null_visitor);
  1831       dfs.init();
  1832 
  1833       typename Graph::Node u = INVALID;
  1834       for (typename Graph::NodeIt n(graph); n != INVALID; ++n) {
  1835         if (!dfs.reached(n)) {
  1836           dfs.addSource(n);
  1837           dfs.start();
  1838           if (u == INVALID) {
  1839             u = n;
  1840           } else {
  1841             typename Graph::Node v = n;
  1842 
  1843             typename Graph::Arc ue = typename Graph::OutArcIt(graph, u);
  1844             typename Graph::Arc ve = typename Graph::OutArcIt(graph, v);
  1845 
  1846             typename Graph::Arc e = graph.direct(graph.addEdge(u, v), true);
  1847 
  1848             if (ue != INVALID) {
  1849               embedding[e] = embedding[ue];
  1850               embedding[ue] = e;
  1851             } else {
  1852               embedding[e] = e;
  1853             }
  1854 
  1855             if (ve != INVALID) {
  1856               embedding[graph.oppositeArc(e)] = embedding[ve];
  1857               embedding[ve] = graph.oppositeArc(e);
  1858             } else {
  1859               embedding[graph.oppositeArc(e)] = graph.oppositeArc(e);
  1860             }
  1861           }
  1862         }
  1863       }
  1864     }
  1865 
  1866     template <typename Graph, typename EmbeddingMap>
  1867     void makeBiNodeConnected(Graph& graph, EmbeddingMap& embedding) {
  1868       typename Graph::template ArcMap<bool> processed(graph);
  1869 
  1870       std::vector<typename Graph::Arc> arcs;
  1871       for (typename Graph::ArcIt e(graph); e != INVALID; ++e) {
  1872         arcs.push_back(e);
  1873       }
  1874 
  1875       IterableBoolMap<Graph, typename Graph::Node> visited(graph, false);
  1876 
  1877       for (int i = 0; i < int(arcs.size()); ++i) {
  1878         typename Graph::Arc pp = arcs[i];
  1879         if (processed[pp]) continue;
  1880 
  1881         typename Graph::Arc e = embedding[graph.oppositeArc(pp)];
  1882         processed[e] = true;
  1883         visited.set(graph.source(e), true);
  1884 
  1885         typename Graph::Arc p = e, l = e;
  1886         e = embedding[graph.oppositeArc(e)];
  1887 
  1888         while (e != l) {
  1889           processed[e] = true;
  1890 
  1891           if (visited[graph.source(e)]) {
  1892 
  1893             typename Graph::Arc n =
  1894               graph.direct(graph.addEdge(graph.source(p),
  1895                                            graph.target(e)), true);
  1896             embedding[n] = p;
  1897             embedding[graph.oppositeArc(pp)] = n;
  1898 
  1899             embedding[graph.oppositeArc(n)] =
  1900               embedding[graph.oppositeArc(e)];
  1901             embedding[graph.oppositeArc(e)] =
  1902               graph.oppositeArc(n);
  1903 
  1904             p = n;
  1905             e = embedding[graph.oppositeArc(n)];
  1906           } else {
  1907             visited.set(graph.source(e), true);
  1908             pp = p;
  1909             p = e;
  1910             e = embedding[graph.oppositeArc(e)];
  1911           }
  1912         }
  1913         visited.setAll(false);
  1914       }
  1915     }
  1916 
  1917 
  1918     template <typename Graph, typename EmbeddingMap>
  1919     void makeMaxPlanar(Graph& graph, EmbeddingMap& embedding) {
  1920 
  1921       typename Graph::template NodeMap<int> degree(graph);
  1922 
  1923       for (typename Graph::NodeIt n(graph); n != INVALID; ++n) {
  1924         degree[n] = countIncEdges(graph, n);
  1925       }
  1926 
  1927       typename Graph::template ArcMap<bool> processed(graph);
  1928       IterableBoolMap<Graph, typename Graph::Node> visited(graph, false);
  1929 
  1930       std::vector<typename Graph::Arc> arcs;
  1931       for (typename Graph::ArcIt e(graph); e != INVALID; ++e) {
  1932         arcs.push_back(e);
  1933       }
  1934 
  1935       for (int i = 0; i < int(arcs.size()); ++i) {
  1936         typename Graph::Arc e = arcs[i];
  1937 
  1938         if (processed[e]) continue;
  1939         processed[e] = true;
  1940 
  1941         typename Graph::Arc mine = e;
  1942         int mind = degree[graph.source(e)];
  1943 
  1944         int face_size = 1;
  1945 
  1946         typename Graph::Arc l = e;
  1947         e = embedding[graph.oppositeArc(e)];
  1948         while (l != e) {
  1949           processed[e] = true;
  1950 
  1951           ++face_size;
  1952 
  1953           if (degree[graph.source(e)] < mind) {
  1954             mine = e;
  1955             mind = degree[graph.source(e)];
  1956           }
  1957 
  1958           e = embedding[graph.oppositeArc(e)];
  1959         }
  1960 
  1961         if (face_size < 4) {
  1962           continue;
  1963         }
  1964 
  1965         typename Graph::Node s = graph.source(mine);
  1966         for (typename Graph::OutArcIt e(graph, s); e != INVALID; ++e) {
  1967           visited.set(graph.target(e), true);
  1968         }
  1969 
  1970         typename Graph::Arc oppe = INVALID;
  1971 
  1972         e = embedding[graph.oppositeArc(mine)];
  1973         e = embedding[graph.oppositeArc(e)];
  1974         while (graph.target(e) != s) {
  1975           if (visited[graph.source(e)]) {
  1976             oppe = e;
  1977             break;
  1978           }
  1979           e = embedding[graph.oppositeArc(e)];
  1980         }
  1981         visited.setAll(false);
  1982 
  1983         if (oppe == INVALID) {
  1984 
  1985           e = embedding[graph.oppositeArc(mine)];
  1986           typename Graph::Arc pn = mine, p = e;
  1987 
  1988           e = embedding[graph.oppositeArc(e)];
  1989           while (graph.target(e) != s) {
  1990             typename Graph::Arc n =
  1991               graph.direct(graph.addEdge(s, graph.source(e)), true);
  1992 
  1993             embedding[n] = pn;
  1994             embedding[graph.oppositeArc(n)] = e;
  1995             embedding[graph.oppositeArc(p)] = graph.oppositeArc(n);
  1996 
  1997             pn = n;
  1998 
  1999             p = e;
  2000             e = embedding[graph.oppositeArc(e)];
  2001           }
  2002 
  2003           embedding[graph.oppositeArc(e)] = pn;
  2004 
  2005         } else {
  2006 
  2007           mine = embedding[graph.oppositeArc(mine)];
  2008           s = graph.source(mine);
  2009           oppe = embedding[graph.oppositeArc(oppe)];
  2010           typename Graph::Node t = graph.source(oppe);
  2011 
  2012           typename Graph::Arc ce = graph.direct(graph.addEdge(s, t), true);
  2013           embedding[ce] = mine;
  2014           embedding[graph.oppositeArc(ce)] = oppe;
  2015 
  2016           typename Graph::Arc pn = ce, p = oppe;
  2017           e = embedding[graph.oppositeArc(oppe)];
  2018           while (graph.target(e) != s) {
  2019             typename Graph::Arc n =
  2020               graph.direct(graph.addEdge(s, graph.source(e)), true);
  2021 
  2022             embedding[n] = pn;
  2023             embedding[graph.oppositeArc(n)] = e;
  2024             embedding[graph.oppositeArc(p)] = graph.oppositeArc(n);
  2025 
  2026             pn = n;
  2027 
  2028             p = e;
  2029             e = embedding[graph.oppositeArc(e)];
  2030 
  2031           }
  2032           embedding[graph.oppositeArc(e)] = pn;
  2033 
  2034           pn = graph.oppositeArc(ce), p = mine;
  2035           e = embedding[graph.oppositeArc(mine)];
  2036           while (graph.target(e) != t) {
  2037             typename Graph::Arc n =
  2038               graph.direct(graph.addEdge(t, graph.source(e)), true);
  2039 
  2040             embedding[n] = pn;
  2041             embedding[graph.oppositeArc(n)] = e;
  2042             embedding[graph.oppositeArc(p)] = graph.oppositeArc(n);
  2043 
  2044             pn = n;
  2045 
  2046             p = e;
  2047             e = embedding[graph.oppositeArc(e)];
  2048 
  2049           }
  2050           embedding[graph.oppositeArc(e)] = pn;
  2051         }
  2052       }
  2053     }
  2054 
  2055   }
  2056 
  2057   /// \ingroup planar
  2058   ///
  2059   /// \brief Schnyder's planar drawing algorithm
  2060   ///
  2061   /// The planar drawing algorithm calculates positions for the nodes
  2062   /// in the plane which coordinates satisfy that if the arcs are
  2063   /// represented with straight lines then they will not intersect
  2064   /// each other.
  2065   ///
  2066   /// Scnyder's algorithm embeds the graph on \c (n-2,n-2) size grid,
  2067   /// i.e. each node will be located in the \c [0,n-2]x[0,n-2] square.
  2068   /// The time complexity of the algorithm is O(n).
  2069   template <typename Graph>
  2070   class PlanarDrawing {
  2071   public:
  2072 
  2073     TEMPLATE_GRAPH_TYPEDEFS(Graph);
  2074 
  2075     /// \brief The point type for store coordinates
  2076     typedef dim2::Point<int> Point;
  2077     /// \brief The map type for store coordinates
  2078     typedef typename Graph::template NodeMap<Point> PointMap;
  2079 
  2080 
  2081     /// \brief Constructor
  2082     ///
  2083     /// Constructor
  2084     /// \pre The graph should be simple, i.e. loop and parallel arc free.
  2085     PlanarDrawing(const Graph& graph)
  2086       : _graph(graph), _point_map(graph) {}
  2087 
  2088   private:
  2089 
  2090     template <typename AuxGraph, typename AuxEmbeddingMap>
  2091     void drawing(const AuxGraph& graph,
  2092                  const AuxEmbeddingMap& next,
  2093                  PointMap& point_map) {
  2094       TEMPLATE_GRAPH_TYPEDEFS(AuxGraph);
  2095 
  2096       typename AuxGraph::template ArcMap<Arc> prev(graph);
  2097 
  2098       for (NodeIt n(graph); n != INVALID; ++n) {
  2099         Arc e = OutArcIt(graph, n);
  2100 
  2101         Arc p = e, l = e;
  2102 
  2103         e = next[e];
  2104         while (e != l) {
  2105           prev[e] = p;
  2106           p = e;
  2107           e = next[e];
  2108         }
  2109         prev[e] = p;
  2110       }
  2111 
  2112       Node anode, bnode, cnode;
  2113 
  2114       {
  2115         Arc e = ArcIt(graph);
  2116         anode = graph.source(e);
  2117         bnode = graph.target(e);
  2118         cnode = graph.target(next[graph.oppositeArc(e)]);
  2119       }
  2120 
  2121       IterableBoolMap<AuxGraph, Node> proper(graph, false);
  2122       typename AuxGraph::template NodeMap<int> conn(graph, -1);
  2123 
  2124       conn[anode] = conn[bnode] = -2;
  2125       {
  2126         for (OutArcIt e(graph, anode); e != INVALID; ++e) {
  2127           Node m = graph.target(e);
  2128           if (conn[m] == -1) {
  2129             conn[m] = 1;
  2130           }
  2131         }
  2132         conn[cnode] = 2;
  2133 
  2134         for (OutArcIt e(graph, bnode); e != INVALID; ++e) {
  2135           Node m = graph.target(e);
  2136           if (conn[m] == -1) {
  2137             conn[m] = 1;
  2138           } else if (conn[m] != -2) {
  2139             conn[m] += 1;
  2140             Arc pe = graph.oppositeArc(e);
  2141             if (conn[graph.target(next[pe])] == -2) {
  2142               conn[m] -= 1;
  2143             }
  2144             if (conn[graph.target(prev[pe])] == -2) {
  2145               conn[m] -= 1;
  2146             }
  2147 
  2148             proper.set(m, conn[m] == 1);
  2149           }
  2150         }
  2151       }
  2152 
  2153 
  2154       typename AuxGraph::template ArcMap<int> angle(graph, -1);
  2155 
  2156       while (proper.trueNum() != 0) {
  2157         Node n = typename IterableBoolMap<AuxGraph, Node>::TrueIt(proper);
  2158         proper.set(n, false);
  2159         conn[n] = -2;
  2160 
  2161         for (OutArcIt e(graph, n); e != INVALID; ++e) {
  2162           Node m = graph.target(e);
  2163           if (conn[m] == -1) {
  2164             conn[m] = 1;
  2165           } else if (conn[m] != -2) {
  2166             conn[m] += 1;
  2167             Arc pe = graph.oppositeArc(e);
  2168             if (conn[graph.target(next[pe])] == -2) {
  2169               conn[m] -= 1;
  2170             }
  2171             if (conn[graph.target(prev[pe])] == -2) {
  2172               conn[m] -= 1;
  2173             }
  2174 
  2175             proper.set(m, conn[m] == 1);
  2176           }
  2177         }
  2178 
  2179         {
  2180           Arc e = OutArcIt(graph, n);
  2181           Arc p = e, l = e;
  2182 
  2183           e = next[e];
  2184           while (e != l) {
  2185 
  2186             if (conn[graph.target(e)] == -2 && conn[graph.target(p)] == -2) {
  2187               Arc f = e;
  2188               angle[f] = 0;
  2189               f = next[graph.oppositeArc(f)];
  2190               angle[f] = 1;
  2191               f = next[graph.oppositeArc(f)];
  2192               angle[f] = 2;
  2193             }
  2194 
  2195             p = e;
  2196             e = next[e];
  2197           }
  2198 
  2199           if (conn[graph.target(e)] == -2 && conn[graph.target(p)] == -2) {
  2200             Arc f = e;
  2201             angle[f] = 0;
  2202             f = next[graph.oppositeArc(f)];
  2203             angle[f] = 1;
  2204             f = next[graph.oppositeArc(f)];
  2205             angle[f] = 2;
  2206           }
  2207         }
  2208       }
  2209 
  2210       typename AuxGraph::template NodeMap<Node> apred(graph, INVALID);
  2211       typename AuxGraph::template NodeMap<Node> bpred(graph, INVALID);
  2212       typename AuxGraph::template NodeMap<Node> cpred(graph, INVALID);
  2213 
  2214       typename AuxGraph::template NodeMap<int> apredid(graph, -1);
  2215       typename AuxGraph::template NodeMap<int> bpredid(graph, -1);
  2216       typename AuxGraph::template NodeMap<int> cpredid(graph, -1);
  2217 
  2218       for (ArcIt e(graph); e != INVALID; ++e) {
  2219         if (angle[e] == angle[next[e]]) {
  2220           switch (angle[e]) {
  2221           case 2:
  2222             apred[graph.target(e)] = graph.source(e);
  2223             apredid[graph.target(e)] = graph.id(graph.source(e));
  2224             break;
  2225           case 1:
  2226             bpred[graph.target(e)] = graph.source(e);
  2227             bpredid[graph.target(e)] = graph.id(graph.source(e));
  2228             break;
  2229           case 0:
  2230             cpred[graph.target(e)] = graph.source(e);
  2231             cpredid[graph.target(e)] = graph.id(graph.source(e));
  2232             break;
  2233           }
  2234         }
  2235       }
  2236 
  2237       cpred[anode] = INVALID;
  2238       cpred[bnode] = INVALID;
  2239 
  2240       std::vector<Node> aorder, border, corder;
  2241 
  2242       {
  2243         typename AuxGraph::template NodeMap<bool> processed(graph, false);
  2244         std::vector<Node> st;
  2245         for (NodeIt n(graph); n != INVALID; ++n) {
  2246           if (!processed[n] && n != bnode && n != cnode) {
  2247             st.push_back(n);
  2248             processed[n] = true;
  2249             Node m = apred[n];
  2250             while (m != INVALID && !processed[m]) {
  2251               st.push_back(m);
  2252               processed[m] = true;
  2253               m = apred[m];
  2254             }
  2255             while (!st.empty()) {
  2256               aorder.push_back(st.back());
  2257               st.pop_back();
  2258             }
  2259           }
  2260         }
  2261       }
  2262 
  2263       {
  2264         typename AuxGraph::template NodeMap<bool> processed(graph, false);
  2265         std::vector<Node> st;
  2266         for (NodeIt n(graph); n != INVALID; ++n) {
  2267           if (!processed[n] && n != cnode && n != anode) {
  2268             st.push_back(n);
  2269             processed[n] = true;
  2270             Node m = bpred[n];
  2271             while (m != INVALID && !processed[m]) {
  2272               st.push_back(m);
  2273               processed[m] = true;
  2274               m = bpred[m];
  2275             }
  2276             while (!st.empty()) {
  2277               border.push_back(st.back());
  2278               st.pop_back();
  2279             }
  2280           }
  2281         }
  2282       }
  2283 
  2284       {
  2285         typename AuxGraph::template NodeMap<bool> processed(graph, false);
  2286         std::vector<Node> st;
  2287         for (NodeIt n(graph); n != INVALID; ++n) {
  2288           if (!processed[n] && n != anode && n != bnode) {
  2289             st.push_back(n);
  2290             processed[n] = true;
  2291             Node m = cpred[n];
  2292             while (m != INVALID && !processed[m]) {
  2293               st.push_back(m);
  2294               processed[m] = true;
  2295               m = cpred[m];
  2296             }
  2297             while (!st.empty()) {
  2298               corder.push_back(st.back());
  2299               st.pop_back();
  2300             }
  2301           }
  2302         }
  2303       }
  2304 
  2305       typename AuxGraph::template NodeMap<int> atree(graph, 0);
  2306       for (int i = aorder.size() - 1; i >= 0; --i) {
  2307         Node n = aorder[i];
  2308         atree[n] = 1;
  2309         for (OutArcIt e(graph, n); e != INVALID; ++e) {
  2310           if (apred[graph.target(e)] == n) {
  2311             atree[n] += atree[graph.target(e)];
  2312           }
  2313         }
  2314       }
  2315 
  2316       typename AuxGraph::template NodeMap<int> btree(graph, 0);
  2317       for (int i = border.size() - 1; i >= 0; --i) {
  2318         Node n = border[i];
  2319         btree[n] = 1;
  2320         for (OutArcIt e(graph, n); e != INVALID; ++e) {
  2321           if (bpred[graph.target(e)] == n) {
  2322             btree[n] += btree[graph.target(e)];
  2323           }
  2324         }
  2325       }
  2326 
  2327       typename AuxGraph::template NodeMap<int> apath(graph, 0);
  2328       apath[bnode] = apath[cnode] = 1;
  2329       typename AuxGraph::template NodeMap<int> apath_btree(graph, 0);
  2330       apath_btree[bnode] = btree[bnode];
  2331       for (int i = 1; i < int(aorder.size()); ++i) {
  2332         Node n = aorder[i];
  2333         apath[n] = apath[apred[n]] + 1;
  2334         apath_btree[n] = btree[n] + apath_btree[apred[n]];
  2335       }
  2336 
  2337       typename AuxGraph::template NodeMap<int> bpath_atree(graph, 0);
  2338       bpath_atree[anode] = atree[anode];
  2339       for (int i = 1; i < int(border.size()); ++i) {
  2340         Node n = border[i];
  2341         bpath_atree[n] = atree[n] + bpath_atree[bpred[n]];
  2342       }
  2343 
  2344       typename AuxGraph::template NodeMap<int> cpath(graph, 0);
  2345       cpath[anode] = cpath[bnode] = 1;
  2346       typename AuxGraph::template NodeMap<int> cpath_atree(graph, 0);
  2347       cpath_atree[anode] = atree[anode];
  2348       typename AuxGraph::template NodeMap<int> cpath_btree(graph, 0);
  2349       cpath_btree[bnode] = btree[bnode];
  2350       for (int i = 1; i < int(corder.size()); ++i) {
  2351         Node n = corder[i];
  2352         cpath[n] = cpath[cpred[n]] + 1;
  2353         cpath_atree[n] = atree[n] + cpath_atree[cpred[n]];
  2354         cpath_btree[n] = btree[n] + cpath_btree[cpred[n]];
  2355       }
  2356 
  2357       typename AuxGraph::template NodeMap<int> third(graph);
  2358       for (NodeIt n(graph); n != INVALID; ++n) {
  2359         point_map[n].x =
  2360           bpath_atree[n] + cpath_atree[n] - atree[n] - cpath[n] + 1;
  2361         point_map[n].y =
  2362           cpath_btree[n] + apath_btree[n] - btree[n] - apath[n] + 1;
  2363       }
  2364 
  2365     }
  2366 
  2367   public:
  2368 
  2369     /// \brief Calculates the node positions
  2370     ///
  2371     /// This function calculates the node positions.
  2372     /// \return %True if the graph is planar.
  2373     bool run() {
  2374       PlanarEmbedding<Graph> pe(_graph);
  2375       if (!pe.run()) return false;
  2376 
  2377       run(pe);
  2378       return true;
  2379     }
  2380 
  2381     /// \brief Calculates the node positions according to a
  2382     /// combinatorical embedding
  2383     ///
  2384     /// This function calculates the node locations. The \c embedding
  2385     /// parameter should contain a valid combinatorical embedding, i.e.
  2386     /// a valid cyclic order of the arcs.
  2387     template <typename EmbeddingMap>
  2388     void run(const EmbeddingMap& embedding) {
  2389       typedef SmartEdgeSet<Graph> AuxGraph;
  2390 
  2391       if (3 * countNodes(_graph) - 6 == countEdges(_graph)) {
  2392         drawing(_graph, embedding, _point_map);
  2393         return;
  2394       }
  2395 
  2396       AuxGraph aux_graph(_graph);
  2397       typename AuxGraph::template ArcMap<typename AuxGraph::Arc>
  2398         aux_embedding(aux_graph);
  2399 
  2400       {
  2401 
  2402         typename Graph::template EdgeMap<typename AuxGraph::Edge>
  2403           ref(_graph);
  2404 
  2405         for (EdgeIt e(_graph); e != INVALID; ++e) {
  2406           ref[e] = aux_graph.addEdge(_graph.u(e), _graph.v(e));
  2407         }
  2408 
  2409         for (EdgeIt e(_graph); e != INVALID; ++e) {
  2410           Arc ee = embedding[_graph.direct(e, true)];
  2411           aux_embedding[aux_graph.direct(ref[e], true)] =
  2412             aux_graph.direct(ref[ee], _graph.direction(ee));
  2413           ee = embedding[_graph.direct(e, false)];
  2414           aux_embedding[aux_graph.direct(ref[e], false)] =
  2415             aux_graph.direct(ref[ee], _graph.direction(ee));
  2416         }
  2417       }
  2418       _planarity_bits::makeConnected(aux_graph, aux_embedding);
  2419       _planarity_bits::makeBiNodeConnected(aux_graph, aux_embedding);
  2420       _planarity_bits::makeMaxPlanar(aux_graph, aux_embedding);
  2421       drawing(aux_graph, aux_embedding, _point_map);
  2422     }
  2423 
  2424     /// \brief The coordinate of the given node
  2425     ///
  2426     /// The coordinate of the given node.
  2427     Point operator[](const Node& node) const {
  2428       return _point_map[node];
  2429     }
  2430 
  2431     /// \brief Returns the grid embedding in a \e NodeMap.
  2432     ///
  2433     /// Returns the grid embedding in a \e NodeMap of \c dim2::Point<int> .
  2434     const PointMap& coords() const {
  2435       return _point_map;
  2436     }
  2437 
  2438   private:
  2439 
  2440     const Graph& _graph;
  2441     PointMap _point_map;
  2442 
  2443   };
  2444 
  2445   namespace _planarity_bits {
  2446 
  2447     template <typename ColorMap>
  2448     class KempeFilter {
  2449     public:
  2450       typedef typename ColorMap::Key Key;
  2451       typedef bool Value;
  2452 
  2453       KempeFilter(const ColorMap& color_map,
  2454                   const typename ColorMap::Value& first,
  2455                   const typename ColorMap::Value& second)
  2456         : _color_map(color_map), _first(first), _second(second) {}
  2457 
  2458       Value operator[](const Key& key) const {
  2459         return _color_map[key] == _first || _color_map[key] == _second;
  2460       }
  2461 
  2462     private:
  2463       const ColorMap& _color_map;
  2464       typename ColorMap::Value _first, _second;
  2465     };
  2466   }
  2467 
  2468   /// \ingroup planar
  2469   ///
  2470   /// \brief Coloring planar graphs
  2471   ///
  2472   /// The graph coloring problem is the coloring of the graph nodes
  2473   /// that there are not adjacent nodes with the same color. The
  2474   /// planar graphs can be always colored with four colors, it is
  2475   /// proved by Appel and Haken and their proofs provide a quadratic
  2476   /// time algorithm for four coloring, but it could not be used to
  2477   /// implement efficient algorithm. The five and six coloring can be
  2478   /// made in linear time, but in this class the five coloring has
  2479   /// quadratic worst case time complexity. The two coloring (if
  2480   /// possible) is solvable with a graph search algorithm and it is
  2481   /// implemented in \ref bipartitePartitions() function in LEMON. To
  2482   /// decide whether the planar graph is three colorable is
  2483   /// NP-complete.
  2484   ///
  2485   /// This class contains member functions for calculate colorings
  2486   /// with five and six colors. The six coloring algorithm is a simple
  2487   /// greedy coloring on the backward minimum outgoing order of nodes.
  2488   /// This order can be computed as in each phase the node with least
  2489   /// outgoing arcs to unprocessed nodes is chosen. This order
  2490   /// guarantees that when a node is chosen for coloring it has at
  2491   /// most five already colored adjacents. The five coloring algorithm
  2492   /// use the same method, but if the greedy approach fails to color
  2493   /// with five colors, i.e. the node has five already different
  2494   /// colored neighbours, it swaps the colors in one of the connected
  2495   /// two colored sets with the Kempe recoloring method.
  2496   template <typename Graph>
  2497   class PlanarColoring {
  2498   public:
  2499 
  2500     TEMPLATE_GRAPH_TYPEDEFS(Graph);
  2501 
  2502     /// \brief The map type for store color indexes
  2503     typedef typename Graph::template NodeMap<int> IndexMap;
  2504     /// \brief The map type for store colors
  2505     typedef ComposeMap<Palette, IndexMap> ColorMap;
  2506 
  2507     /// \brief Constructor
  2508     ///
  2509     /// Constructor
  2510     /// \pre The graph should be simple, i.e. loop and parallel arc free.
  2511     PlanarColoring(const Graph& graph)
  2512       : _graph(graph), _color_map(graph), _palette(0) {
  2513       _palette.add(Color(1,0,0));
  2514       _palette.add(Color(0,1,0));
  2515       _palette.add(Color(0,0,1));
  2516       _palette.add(Color(1,1,0));
  2517       _palette.add(Color(1,0,1));
  2518       _palette.add(Color(0,1,1));
  2519     }
  2520 
  2521     /// \brief Returns the \e NodeMap of color indexes
  2522     ///
  2523     /// Returns the \e NodeMap of color indexes. The values are in the
  2524     /// range \c [0..4] or \c [0..5] according to the coloring method.
  2525     IndexMap colorIndexMap() const {
  2526       return _color_map;
  2527     }
  2528 
  2529     /// \brief Returns the \e NodeMap of colors
  2530     ///
  2531     /// Returns the \e NodeMap of colors. The values are five or six
  2532     /// distinct \ref lemon::Color "colors".
  2533     ColorMap colorMap() const {
  2534       return composeMap(_palette, _color_map);
  2535     }
  2536 
  2537     /// \brief Returns the color index of the node
  2538     ///
  2539     /// Returns the color index of the node. The values are in the
  2540     /// range \c [0..4] or \c [0..5] according to the coloring method.
  2541     int colorIndex(const Node& node) const {
  2542       return _color_map[node];
  2543     }
  2544 
  2545     /// \brief Returns the color of the node
  2546     ///
  2547     /// Returns the color of the node. The values are five or six
  2548     /// distinct \ref lemon::Color "colors".
  2549     Color color(const Node& node) const {
  2550       return _palette[_color_map[node]];
  2551     }
  2552 
  2553 
  2554     /// \brief Calculates a coloring with at most six colors
  2555     ///
  2556     /// This function calculates a coloring with at most six colors. The time
  2557     /// complexity of this variant is linear in the size of the graph.
  2558     /// \return %True when the algorithm could color the graph with six color.
  2559     /// If the algorithm fails, then the graph could not be planar.
  2560     /// \note This function can return true if the graph is not
  2561     /// planar but it can be colored with 6 colors.
  2562     bool runSixColoring() {
  2563 
  2564       typename Graph::template NodeMap<int> heap_index(_graph, -1);
  2565       BucketHeap<typename Graph::template NodeMap<int> > heap(heap_index);
  2566 
  2567       for (NodeIt n(_graph); n != INVALID; ++n) {
  2568         _color_map[n] = -2;
  2569         heap.push(n, countOutArcs(_graph, n));
  2570       }
  2571 
  2572       std::vector<Node> order;
  2573 
  2574       while (!heap.empty()) {
  2575         Node n = heap.top();
  2576         heap.pop();
  2577         _color_map[n] = -1;
  2578         order.push_back(n);
  2579         for (OutArcIt e(_graph, n); e != INVALID; ++e) {
  2580           Node t = _graph.runningNode(e);
  2581           if (_color_map[t] == -2) {
  2582             heap.decrease(t, heap[t] - 1);
  2583           }
  2584         }
  2585       }
  2586 
  2587       for (int i = order.size() - 1; i >= 0; --i) {
  2588         std::vector<bool> forbidden(6, false);
  2589         for (OutArcIt e(_graph, order[i]); e != INVALID; ++e) {
  2590           Node t = _graph.runningNode(e);
  2591           if (_color_map[t] != -1) {
  2592             forbidden[_color_map[t]] = true;
  2593           }
  2594         }
  2595                for (int k = 0; k < 6; ++k) {
  2596           if (!forbidden[k]) {
  2597             _color_map[order[i]] = k;
  2598             break;
  2599           }
  2600         }
  2601         if (_color_map[order[i]] == -1) {
  2602           return false;
  2603         }
  2604       }
  2605       return true;
  2606     }
  2607 
  2608   private:
  2609 
  2610     bool recolor(const Node& u, const Node& v) {
  2611       int ucolor = _color_map[u];
  2612       int vcolor = _color_map[v];
  2613       typedef _planarity_bits::KempeFilter<IndexMap> KempeFilter;
  2614       KempeFilter filter(_color_map, ucolor, vcolor);
  2615 
  2616       typedef FilterNodes<const Graph, const KempeFilter> KempeGraph;
  2617       KempeGraph kempe_graph(_graph, filter);
  2618 
  2619       std::vector<Node> comp;
  2620       Bfs<KempeGraph> bfs(kempe_graph);
  2621       bfs.init();
  2622       bfs.addSource(u);
  2623       while (!bfs.emptyQueue()) {
  2624         Node n = bfs.nextNode();
  2625         if (n == v) return false;
  2626         comp.push_back(n);
  2627         bfs.processNextNode();
  2628       }
  2629 
  2630       int scolor = ucolor + vcolor;
  2631       for (int i = 0; i < static_cast<int>(comp.size()); ++i) {
  2632         _color_map[comp[i]] = scolor - _color_map[comp[i]];
  2633       }
  2634 
  2635       return true;
  2636     }
  2637 
  2638     template <typename EmbeddingMap>
  2639     void kempeRecoloring(const Node& node, const EmbeddingMap& embedding) {
  2640       std::vector<Node> nodes;
  2641       nodes.reserve(4);
  2642 
  2643       for (Arc e = OutArcIt(_graph, node); e != INVALID; e = embedding[e]) {
  2644         Node t = _graph.target(e);
  2645         if (_color_map[t] != -1) {
  2646           nodes.push_back(t);
  2647           if (nodes.size() == 4) break;
  2648         }
  2649       }
  2650 
  2651       int color = _color_map[nodes[0]];
  2652       if (recolor(nodes[0], nodes[2])) {
  2653         _color_map[node] = color;
  2654       } else {
  2655         color = _color_map[nodes[1]];
  2656         recolor(nodes[1], nodes[3]);
  2657         _color_map[node] = color;
  2658       }
  2659     }
  2660 
  2661   public:
  2662 
  2663     /// \brief Calculates a coloring with at most five colors
  2664     ///
  2665     /// This function calculates a coloring with at most five
  2666     /// colors. The worst case time complexity of this variant is
  2667     /// quadratic in the size of the graph.
  2668     template <typename EmbeddingMap>
  2669     void runFiveColoring(const EmbeddingMap& embedding) {
  2670 
  2671       typename Graph::template NodeMap<int> heap_index(_graph, -1);
  2672       BucketHeap<typename Graph::template NodeMap<int> > heap(heap_index);
  2673 
  2674       for (NodeIt n(_graph); n != INVALID; ++n) {
  2675         _color_map[n] = -2;
  2676         heap.push(n, countOutArcs(_graph, n));
  2677       }
  2678 
  2679       std::vector<Node> order;
  2680 
  2681       while (!heap.empty()) {
  2682         Node n = heap.top();
  2683         heap.pop();
  2684         _color_map[n] = -1;
  2685         order.push_back(n);
  2686         for (OutArcIt e(_graph, n); e != INVALID; ++e) {
  2687           Node t = _graph.runningNode(e);
  2688           if (_color_map[t] == -2) {
  2689             heap.decrease(t, heap[t] - 1);
  2690           }
  2691         }
  2692       }
  2693 
  2694       for (int i = order.size() - 1; i >= 0; --i) {
  2695         std::vector<bool> forbidden(5, false);
  2696         for (OutArcIt e(_graph, order[i]); e != INVALID; ++e) {
  2697           Node t = _graph.runningNode(e);
  2698           if (_color_map[t] != -1) {
  2699             forbidden[_color_map[t]] = true;
  2700           }
  2701         }
  2702         for (int k = 0; k < 5; ++k) {
  2703           if (!forbidden[k]) {
  2704             _color_map[order[i]] = k;
  2705             break;
  2706           }
  2707         }
  2708         if (_color_map[order[i]] == -1) {
  2709           kempeRecoloring(order[i], embedding);
  2710         }
  2711       }
  2712     }
  2713 
  2714     /// \brief Calculates a coloring with at most five colors
  2715     ///
  2716     /// This function calculates a coloring with at most five
  2717     /// colors. The worst case time complexity of this variant is
  2718     /// quadratic in the size of the graph.
  2719     /// \return %True when the graph is planar.
  2720     bool runFiveColoring() {
  2721       PlanarEmbedding<Graph> pe(_graph);
  2722       if (!pe.run()) return false;
  2723 
  2724       runFiveColoring(pe.embeddingMap());
  2725       return true;
  2726     }
  2727 
  2728   private:
  2729 
  2730     const Graph& _graph;
  2731     IndexMap _color_map;
  2732     Palette _palette;
  2733   };
  2734 
  2735 }
  2736 
  2737 #endif