diff -r 994c7df296c9 -r 1b89e29c9fc7 lemon/planarity.h
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/lemon/planarity.h	Thu Dec 10 17:18:25 2009 +0100
@@ -0,0 +1,2737 @@
+/* -*- mode: C++; indent-tabs-mode: nil; -*-
+ *
+ * This file is a part of LEMON, a generic C++ optimization library.
+ *
+ * Copyright (C) 2003-2009
+ * Egervary Jeno Kombinatorikus Optimalizalasi Kutatocsoport
+ * (Egervary Research Group on Combinatorial Optimization, EGRES).
+ *
+ * Permission to use, modify and distribute this software is granted
+ * provided that this copyright notice appears in all copies. For
+ * precise terms see the accompanying LICENSE file.
+ *
+ * This software is provided "AS IS" with no warranty of any kind,
+ * express or implied, and with no claim as to its suitability for any
+ * purpose.
+ *
+ */
+
+#ifndef LEMON_PLANARITY_H
+#define LEMON_PLANARITY_H
+
+/// \ingroup planar
+/// \file
+/// \brief Planarity checking, embedding, drawing and coloring
+
+#include <vector>
+#include <list>
+
+#include <lemon/dfs.h>
+#include <lemon/bfs.h>
+#include <lemon/radix_sort.h>
+#include <lemon/maps.h>
+#include <lemon/path.h>
+#include <lemon/bucket_heap.h>
+#include <lemon/adaptors.h>
+#include <lemon/edge_set.h>
+#include <lemon/color.h>
+#include <lemon/dim2.h>
+
+namespace lemon {
+
+  namespace _planarity_bits {
+
+    template <typename Graph>
+    struct PlanarityVisitor : DfsVisitor<Graph> {
+
+      TEMPLATE_GRAPH_TYPEDEFS(Graph);
+
+      typedef typename Graph::template NodeMap<Arc> PredMap;
+
+      typedef typename Graph::template EdgeMap<bool> TreeMap;
+
+      typedef typename Graph::template NodeMap<int> OrderMap;
+      typedef std::vector<Node> OrderList;
+
+      typedef typename Graph::template NodeMap<int> LowMap;
+      typedef typename Graph::template NodeMap<int> AncestorMap;
+
+      PlanarityVisitor(const Graph& graph,
+                       PredMap& pred_map, TreeMap& tree_map,
+                       OrderMap& order_map, OrderList& order_list,
+                       AncestorMap& ancestor_map, LowMap& low_map)
+        : _graph(graph), _pred_map(pred_map), _tree_map(tree_map),
+          _order_map(order_map), _order_list(order_list),
+          _ancestor_map(ancestor_map), _low_map(low_map) {}
+
+      void reach(const Node& node) {
+        _order_map[node] = _order_list.size();
+        _low_map[node] = _order_list.size();
+        _ancestor_map[node] = _order_list.size();
+        _order_list.push_back(node);
+      }
+
+      void discover(const Arc& arc) {
+        Node source = _graph.source(arc);
+        Node target = _graph.target(arc);
+
+        _tree_map[arc] = true;
+        _pred_map[target] = arc;
+      }
+
+      void examine(const Arc& arc) {
+        Node source = _graph.source(arc);
+        Node target = _graph.target(arc);
+
+        if (_order_map[target] < _order_map[source] && !_tree_map[arc]) {
+          if (_low_map[source] > _order_map[target]) {
+            _low_map[source] = _order_map[target];
+          }
+          if (_ancestor_map[source] > _order_map[target]) {
+            _ancestor_map[source] = _order_map[target];
+          }
+        }
+      }
+
+      void backtrack(const Arc& arc) {
+        Node source = _graph.source(arc);
+        Node target = _graph.target(arc);
+
+        if (_low_map[source] > _low_map[target]) {
+          _low_map[source] = _low_map[target];
+        }
+      }
+
+      const Graph& _graph;
+      PredMap& _pred_map;
+      TreeMap& _tree_map;
+      OrderMap& _order_map;
+      OrderList& _order_list;
+      AncestorMap& _ancestor_map;
+      LowMap& _low_map;
+    };
+
+    template <typename Graph, bool embedding = true>
+    struct NodeDataNode {
+      int prev, next;
+      int visited;
+      typename Graph::Arc first;
+      bool inverted;
+    };
+
+    template <typename Graph>
+    struct NodeDataNode<Graph, false> {
+      int prev, next;
+      int visited;
+    };
+
+    template <typename Graph>
+    struct ChildListNode {
+      typedef typename Graph::Node Node;
+      Node first;
+      Node prev, next;
+    };
+
+    template <typename Graph>
+    struct ArcListNode {
+      typename Graph::Arc prev, next;
+    };
+
+    template <typename Graph>
+    class PlanarityChecking {
+    private:
+      
+      TEMPLATE_GRAPH_TYPEDEFS(Graph);
+
+      const Graph& _graph;
+
+    private:
+      
+      typedef typename Graph::template NodeMap<Arc> PredMap;
+      
+      typedef typename Graph::template EdgeMap<bool> TreeMap;
+      
+      typedef typename Graph::template NodeMap<int> OrderMap;
+      typedef std::vector<Node> OrderList;
+
+      typedef typename Graph::template NodeMap<int> LowMap;
+      typedef typename Graph::template NodeMap<int> AncestorMap;
+
+      typedef _planarity_bits::NodeDataNode<Graph> NodeDataNode;
+      typedef std::vector<NodeDataNode> NodeData;
+
+      typedef _planarity_bits::ChildListNode<Graph> ChildListNode;
+      typedef typename Graph::template NodeMap<ChildListNode> ChildLists;
+
+      typedef typename Graph::template NodeMap<std::list<int> > MergeRoots;
+
+      typedef typename Graph::template NodeMap<bool> EmbedArc;
+
+    public:
+
+      PlanarityChecking(const Graph& graph) : _graph(graph) {}
+
+      bool run() {
+        typedef _planarity_bits::PlanarityVisitor<Graph> Visitor;
+
+        PredMap pred_map(_graph, INVALID);
+        TreeMap tree_map(_graph, false);
+
+        OrderMap order_map(_graph, -1);
+        OrderList order_list;
+
+        AncestorMap ancestor_map(_graph, -1);
+        LowMap low_map(_graph, -1);
+
+        Visitor visitor(_graph, pred_map, tree_map,
+                        order_map, order_list, ancestor_map, low_map);
+        DfsVisit<Graph, Visitor> visit(_graph, visitor);
+        visit.run();
+
+        ChildLists child_lists(_graph);
+        createChildLists(tree_map, order_map, low_map, child_lists);
+
+        NodeData node_data(2 * order_list.size());
+
+        EmbedArc embed_arc(_graph, false);
+
+        MergeRoots merge_roots(_graph);
+
+        for (int i = order_list.size() - 1; i >= 0; --i) {
+
+          Node node = order_list[i];
+
+          Node source = node;
+          for (OutArcIt e(_graph, node); e != INVALID; ++e) {
+            Node target = _graph.target(e);
+
+            if (order_map[source] < order_map[target] && tree_map[e]) {
+              initFace(target, node_data, order_map, order_list);
+            }
+          }
+
+          for (OutArcIt e(_graph, node); e != INVALID; ++e) {
+            Node target = _graph.target(e);
+
+            if (order_map[source] < order_map[target] && !tree_map[e]) {
+              embed_arc[target] = true;
+              walkUp(target, source, i, pred_map, low_map,
+                     order_map, order_list, node_data, merge_roots);
+            }
+          }
+
+          for (typename MergeRoots::Value::iterator it =
+                 merge_roots[node].begin(); 
+               it != merge_roots[node].end(); ++it) {
+            int rn = *it;
+            walkDown(rn, i, node_data, order_list, child_lists,
+                     ancestor_map, low_map, embed_arc, merge_roots);
+          }
+          merge_roots[node].clear();
+
+          for (OutArcIt e(_graph, node); e != INVALID; ++e) {
+            Node target = _graph.target(e);
+
+            if (order_map[source] < order_map[target] && !tree_map[e]) {
+              if (embed_arc[target]) {
+                return false;
+              }
+            }
+          }
+        }
+
+        return true;
+      }
+
+    private:
+
+      void createChildLists(const TreeMap& tree_map, const OrderMap& order_map,
+                            const LowMap& low_map, ChildLists& child_lists) {
+
+        for (NodeIt n(_graph); n != INVALID; ++n) {
+          Node source = n;
+
+          std::vector<Node> targets;
+          for (OutArcIt e(_graph, n); e != INVALID; ++e) {
+            Node target = _graph.target(e);
+
+            if (order_map[source] < order_map[target] && tree_map[e]) {
+              targets.push_back(target);
+            }
+          }
+
+          if (targets.size() == 0) {
+            child_lists[source].first = INVALID;
+          } else if (targets.size() == 1) {
+            child_lists[source].first = targets[0];
+            child_lists[targets[0]].prev = INVALID;
+            child_lists[targets[0]].next = INVALID;
+          } else {
+            radixSort(targets.begin(), targets.end(), mapToFunctor(low_map));
+            for (int i = 1; i < int(targets.size()); ++i) {
+              child_lists[targets[i]].prev = targets[i - 1];
+              child_lists[targets[i - 1]].next = targets[i];
+            }
+            child_lists[targets.back()].next = INVALID;
+            child_lists[targets.front()].prev = INVALID;
+            child_lists[source].first = targets.front();
+          }
+        }
+      }
+
+      void walkUp(const Node& node, Node root, int rorder,
+                  const PredMap& pred_map, const LowMap& low_map,
+                  const OrderMap& order_map, const OrderList& order_list,
+                  NodeData& node_data, MergeRoots& merge_roots) {
+
+        int na, nb;
+        bool da, db;
+
+        na = nb = order_map[node];
+        da = true; db = false;
+
+        while (true) {
+
+          if (node_data[na].visited == rorder) break;
+          if (node_data[nb].visited == rorder) break;
+
+          node_data[na].visited = rorder;
+          node_data[nb].visited = rorder;
+
+          int rn = -1;
+
+          if (na >= int(order_list.size())) {
+            rn = na;
+          } else if (nb >= int(order_list.size())) {
+            rn = nb;
+          }
+
+          if (rn == -1) {
+            int nn;
+
+            nn = da ? node_data[na].prev : node_data[na].next;
+            da = node_data[nn].prev != na;
+            na = nn;
+
+            nn = db ? node_data[nb].prev : node_data[nb].next;
+            db = node_data[nn].prev != nb;
+            nb = nn;
+
+          } else {
+
+            Node rep = order_list[rn - order_list.size()];
+            Node parent = _graph.source(pred_map[rep]);
+
+            if (low_map[rep] < rorder) {
+              merge_roots[parent].push_back(rn);
+            } else {
+              merge_roots[parent].push_front(rn);
+            }
+
+            if (parent != root) {
+              na = nb = order_map[parent];
+              da = true; db = false;
+            } else {
+              break;
+            }
+          }
+        }
+      }
+
+      void walkDown(int rn, int rorder, NodeData& node_data,
+                    OrderList& order_list, ChildLists& child_lists,
+                    AncestorMap& ancestor_map, LowMap& low_map,
+                    EmbedArc& embed_arc, MergeRoots& merge_roots) {
+
+        std::vector<std::pair<int, bool> > merge_stack;
+
+        for (int di = 0; di < 2; ++di) {
+          bool rd = di == 0;
+          int pn = rn;
+          int n = rd ? node_data[rn].next : node_data[rn].prev;
+
+          while (n != rn) {
+
+            Node node = order_list[n];
+
+            if (embed_arc[node]) {
+
+              // Merging components on the critical path
+              while (!merge_stack.empty()) {
+
+                // Component root
+                int cn = merge_stack.back().first;
+                bool cd = merge_stack.back().second;
+                merge_stack.pop_back();
+
+                // Parent of component
+                int dn = merge_stack.back().first;
+                bool dd = merge_stack.back().second;
+                merge_stack.pop_back();
+
+                Node parent = order_list[dn];
+
+                // Erasing from merge_roots
+                merge_roots[parent].pop_front();
+
+                Node child = order_list[cn - order_list.size()];
+
+                // Erasing from child_lists
+                if (child_lists[child].prev != INVALID) {
+                  child_lists[child_lists[child].prev].next =
+                    child_lists[child].next;
+                } else {
+                  child_lists[parent].first = child_lists[child].next;
+                }
+
+                if (child_lists[child].next != INVALID) {
+                  child_lists[child_lists[child].next].prev =
+                    child_lists[child].prev;
+                }
+
+                // Merging external faces
+                {
+                  int en = cn;
+                  cn = cd ? node_data[cn].prev : node_data[cn].next;
+                  cd = node_data[cn].next == en;
+
+                }
+
+                if (cd) node_data[cn].next = dn; else node_data[cn].prev = dn;
+                if (dd) node_data[dn].prev = cn; else node_data[dn].next = cn;
+
+              }
+
+              bool d = pn == node_data[n].prev;
+
+              if (node_data[n].prev == node_data[n].next &&
+                  node_data[n].inverted) {
+                d = !d;
+              }
+
+              // Embedding arc into external face
+              if (rd) node_data[rn].next = n; else node_data[rn].prev = n;
+              if (d) node_data[n].prev = rn; else node_data[n].next = rn;
+              pn = rn;
+
+              embed_arc[order_list[n]] = false;
+            }
+
+            if (!merge_roots[node].empty()) {
+
+              bool d = pn == node_data[n].prev;
+
+              merge_stack.push_back(std::make_pair(n, d));
+
+              int rn = merge_roots[node].front();
+
+              int xn = node_data[rn].next;
+              Node xnode = order_list[xn];
+
+              int yn = node_data[rn].prev;
+              Node ynode = order_list[yn];
+
+              bool rd;
+              if (!external(xnode, rorder, child_lists, 
+                            ancestor_map, low_map)) {
+                rd = true;
+              } else if (!external(ynode, rorder, child_lists,
+                                   ancestor_map, low_map)) {
+                rd = false;
+              } else if (pertinent(xnode, embed_arc, merge_roots)) {
+                rd = true;
+              } else {
+                rd = false;
+              }
+
+              merge_stack.push_back(std::make_pair(rn, rd));
+
+              pn = rn;
+              n = rd ? xn : yn;
+
+            } else if (!external(node, rorder, child_lists,
+                                 ancestor_map, low_map)) {
+              int nn = (node_data[n].next != pn ?
+                        node_data[n].next : node_data[n].prev);
+
+              bool nd = n == node_data[nn].prev;
+
+              if (nd) node_data[nn].prev = pn;
+              else node_data[nn].next = pn;
+
+              if (n == node_data[pn].prev) node_data[pn].prev = nn;
+              else node_data[pn].next = nn;
+
+              node_data[nn].inverted =
+                (node_data[nn].prev == node_data[nn].next && nd != rd);
+
+              n = nn;
+            }
+            else break;
+
+          }
+
+          if (!merge_stack.empty() || n == rn) {
+            break;
+          }
+        }
+      }
+
+      void initFace(const Node& node, NodeData& node_data,
+                    const OrderMap& order_map, const OrderList& order_list) {
+        int n = order_map[node];
+        int rn = n + order_list.size();
+
+        node_data[n].next = node_data[n].prev = rn;
+        node_data[rn].next = node_data[rn].prev = n;
+
+        node_data[n].visited = order_list.size();
+        node_data[rn].visited = order_list.size();
+
+      }
+
+      bool external(const Node& node, int rorder,
+                    ChildLists& child_lists, AncestorMap& ancestor_map,
+                    LowMap& low_map) {
+        Node child = child_lists[node].first;
+
+        if (child != INVALID) {
+          if (low_map[child] < rorder) return true;
+        }
+
+        if (ancestor_map[node] < rorder) return true;
+
+        return false;
+      }
+
+      bool pertinent(const Node& node, const EmbedArc& embed_arc,
+                     const MergeRoots& merge_roots) {
+        return !merge_roots[node].empty() || embed_arc[node];
+      }
+
+    };
+
+  }
+
+  /// \ingroup planar
+  ///
+  /// \brief Planarity checking of an undirected simple graph
+  ///
+  /// This function implements the Boyer-Myrvold algorithm for
+  /// planarity checking of an undirected graph. It is a simplified
+  /// version of the PlanarEmbedding algorithm class because neither
+  /// the embedding nor the kuratowski subdivisons are not computed.
+  template <typename GR>
+  bool checkPlanarity(const GR& graph) {
+    _planarity_bits::PlanarityChecking<GR> pc(graph);
+    return pc.run();
+  }
+
+  /// \ingroup planar
+  ///
+  /// \brief Planar embedding of an undirected simple graph
+  ///
+  /// This class implements the Boyer-Myrvold algorithm for planar
+  /// embedding of an undirected graph. The planar embedding is an
+  /// ordering of the outgoing edges of the nodes, which is a possible
+  /// configuration to draw the graph in the plane. If there is not
+  /// such ordering then the graph contains a \f$ K_5 \f$ (full graph
+  /// with 5 nodes) or a \f$ K_{3,3} \f$ (complete bipartite graph on
+  /// 3 ANode and 3 BNode) subdivision.
+  ///
+  /// The current implementation calculates either an embedding or a
+  /// Kuratowski subdivision. The running time of the algorithm is 
+  /// \f$ O(n) \f$.
+  template <typename Graph>
+  class PlanarEmbedding {
+  private:
+
+    TEMPLATE_GRAPH_TYPEDEFS(Graph);
+
+    const Graph& _graph;
+    typename Graph::template ArcMap<Arc> _embedding;
+
+    typename Graph::template EdgeMap<bool> _kuratowski;
+
+  private:
+
+    typedef typename Graph::template NodeMap<Arc> PredMap;
+
+    typedef typename Graph::template EdgeMap<bool> TreeMap;
+
+    typedef typename Graph::template NodeMap<int> OrderMap;
+    typedef std::vector<Node> OrderList;
+
+    typedef typename Graph::template NodeMap<int> LowMap;
+    typedef typename Graph::template NodeMap<int> AncestorMap;
+
+    typedef _planarity_bits::NodeDataNode<Graph> NodeDataNode;
+    typedef std::vector<NodeDataNode> NodeData;
+
+    typedef _planarity_bits::ChildListNode<Graph> ChildListNode;
+    typedef typename Graph::template NodeMap<ChildListNode> ChildLists;
+
+    typedef typename Graph::template NodeMap<std::list<int> > MergeRoots;
+
+    typedef typename Graph::template NodeMap<Arc> EmbedArc;
+
+    typedef _planarity_bits::ArcListNode<Graph> ArcListNode;
+    typedef typename Graph::template ArcMap<ArcListNode> ArcLists;
+
+    typedef typename Graph::template NodeMap<bool> FlipMap;
+
+    typedef typename Graph::template NodeMap<int> TypeMap;
+
+    enum IsolatorNodeType {
+      HIGHX = 6, LOWX = 7,
+      HIGHY = 8, LOWY = 9,
+      ROOT = 10, PERTINENT = 11,
+      INTERNAL = 12
+    };
+
+  public:
+
+    /// \brief The map for store of embedding
+    typedef typename Graph::template ArcMap<Arc> EmbeddingMap;
+
+    /// \brief Constructor
+    ///
+    /// \note The graph should be simple, i.e. parallel and loop arc
+    /// free.
+    PlanarEmbedding(const Graph& graph)
+      : _graph(graph), _embedding(_graph), _kuratowski(graph, false) {}
+
+    /// \brief Runs the algorithm.
+    ///
+    /// Runs the algorithm.
+    /// \param kuratowski If the parameter is false, then the
+    /// algorithm does not compute a Kuratowski subdivision.
+    ///\return %True when the graph is planar.
+    bool run(bool kuratowski = true) {
+      typedef _planarity_bits::PlanarityVisitor<Graph> Visitor;
+
+      PredMap pred_map(_graph, INVALID);
+      TreeMap tree_map(_graph, false);
+
+      OrderMap order_map(_graph, -1);
+      OrderList order_list;
+
+      AncestorMap ancestor_map(_graph, -1);
+      LowMap low_map(_graph, -1);
+
+      Visitor visitor(_graph, pred_map, tree_map,
+                      order_map, order_list, ancestor_map, low_map);
+      DfsVisit<Graph, Visitor> visit(_graph, visitor);
+      visit.run();
+
+      ChildLists child_lists(_graph);
+      createChildLists(tree_map, order_map, low_map, child_lists);
+
+      NodeData node_data(2 * order_list.size());
+
+      EmbedArc embed_arc(_graph, INVALID);
+
+      MergeRoots merge_roots(_graph);
+
+      ArcLists arc_lists(_graph);
+
+      FlipMap flip_map(_graph, false);
+
+      for (int i = order_list.size() - 1; i >= 0; --i) {
+
+        Node node = order_list[i];
+
+        node_data[i].first = INVALID;
+
+        Node source = node;
+        for (OutArcIt e(_graph, node); e != INVALID; ++e) {
+          Node target = _graph.target(e);
+
+          if (order_map[source] < order_map[target] && tree_map[e]) {
+            initFace(target, arc_lists, node_data,
+                     pred_map, order_map, order_list);
+          }
+        }
+
+        for (OutArcIt e(_graph, node); e != INVALID; ++e) {
+          Node target = _graph.target(e);
+
+          if (order_map[source] < order_map[target] && !tree_map[e]) {
+            embed_arc[target] = e;
+            walkUp(target, source, i, pred_map, low_map,
+                   order_map, order_list, node_data, merge_roots);
+          }
+        }
+
+        for (typename MergeRoots::Value::iterator it =
+               merge_roots[node].begin(); it != merge_roots[node].end(); ++it) {
+          int rn = *it;
+          walkDown(rn, i, node_data, arc_lists, flip_map, order_list,
+                   child_lists, ancestor_map, low_map, embed_arc, merge_roots);
+        }
+        merge_roots[node].clear();
+
+        for (OutArcIt e(_graph, node); e != INVALID; ++e) {
+          Node target = _graph.target(e);
+
+          if (order_map[source] < order_map[target] && !tree_map[e]) {
+            if (embed_arc[target] != INVALID) {
+              if (kuratowski) {
+                isolateKuratowski(e, node_data, arc_lists, flip_map,
+                                  order_map, order_list, pred_map, child_lists,
+                                  ancestor_map, low_map,
+                                  embed_arc, merge_roots);
+              }
+              return false;
+            }
+          }
+        }
+      }
+
+      for (int i = 0; i < int(order_list.size()); ++i) {
+
+        mergeRemainingFaces(order_list[i], node_data, order_list, order_map,
+                            child_lists, arc_lists);
+        storeEmbedding(order_list[i], node_data, order_map, pred_map,
+                       arc_lists, flip_map);
+      }
+
+      return true;
+    }
+
+    /// \brief Gives back the successor of an arc
+    ///
+    /// Gives back the successor of an arc. This function makes
+    /// possible to query the cyclic order of the outgoing arcs from
+    /// a node.
+    Arc next(const Arc& arc) const {
+      return _embedding[arc];
+    }
+
+    /// \brief Gives back the calculated embedding map
+    ///
+    /// The returned map contains the successor of each arc in the
+    /// graph.
+    const EmbeddingMap& embeddingMap() const {
+      return _embedding;
+    }
+
+    /// \brief Gives back true if the undirected arc is in the
+    /// kuratowski subdivision
+    ///
+    /// Gives back true if the undirected arc is in the kuratowski
+    /// subdivision
+    /// \note The \c run() had to be called with true value.
+    bool kuratowski(const Edge& edge) {
+      return _kuratowski[edge];
+    }
+
+  private:
+
+    void createChildLists(const TreeMap& tree_map, const OrderMap& order_map,
+                          const LowMap& low_map, ChildLists& child_lists) {
+
+      for (NodeIt n(_graph); n != INVALID; ++n) {
+        Node source = n;
+
+        std::vector<Node> targets;
+        for (OutArcIt e(_graph, n); e != INVALID; ++e) {
+          Node target = _graph.target(e);
+
+          if (order_map[source] < order_map[target] && tree_map[e]) {
+            targets.push_back(target);
+          }
+        }
+
+        if (targets.size() == 0) {
+          child_lists[source].first = INVALID;
+        } else if (targets.size() == 1) {
+          child_lists[source].first = targets[0];
+          child_lists[targets[0]].prev = INVALID;
+          child_lists[targets[0]].next = INVALID;
+        } else {
+          radixSort(targets.begin(), targets.end(), mapToFunctor(low_map));
+          for (int i = 1; i < int(targets.size()); ++i) {
+            child_lists[targets[i]].prev = targets[i - 1];
+            child_lists[targets[i - 1]].next = targets[i];
+          }
+          child_lists[targets.back()].next = INVALID;
+          child_lists[targets.front()].prev = INVALID;
+          child_lists[source].first = targets.front();
+        }
+      }
+    }
+
+    void walkUp(const Node& node, Node root, int rorder,
+                const PredMap& pred_map, const LowMap& low_map,
+                const OrderMap& order_map, const OrderList& order_list,
+                NodeData& node_data, MergeRoots& merge_roots) {
+
+      int na, nb;
+      bool da, db;
+
+      na = nb = order_map[node];
+      da = true; db = false;
+
+      while (true) {
+
+        if (node_data[na].visited == rorder) break;
+        if (node_data[nb].visited == rorder) break;
+
+        node_data[na].visited = rorder;
+        node_data[nb].visited = rorder;
+
+        int rn = -1;
+
+        if (na >= int(order_list.size())) {
+          rn = na;
+        } else if (nb >= int(order_list.size())) {
+          rn = nb;
+        }
+
+        if (rn == -1) {
+          int nn;
+
+          nn = da ? node_data[na].prev : node_data[na].next;
+          da = node_data[nn].prev != na;
+          na = nn;
+
+          nn = db ? node_data[nb].prev : node_data[nb].next;
+          db = node_data[nn].prev != nb;
+          nb = nn;
+
+        } else {
+
+          Node rep = order_list[rn - order_list.size()];
+          Node parent = _graph.source(pred_map[rep]);
+
+          if (low_map[rep] < rorder) {
+            merge_roots[parent].push_back(rn);
+          } else {
+            merge_roots[parent].push_front(rn);
+          }
+
+          if (parent != root) {
+            na = nb = order_map[parent];
+            da = true; db = false;
+          } else {
+            break;
+          }
+        }
+      }
+    }
+
+    void walkDown(int rn, int rorder, NodeData& node_data,
+                  ArcLists& arc_lists, FlipMap& flip_map,
+                  OrderList& order_list, ChildLists& child_lists,
+                  AncestorMap& ancestor_map, LowMap& low_map,
+                  EmbedArc& embed_arc, MergeRoots& merge_roots) {
+
+      std::vector<std::pair<int, bool> > merge_stack;
+
+      for (int di = 0; di < 2; ++di) {
+        bool rd = di == 0;
+        int pn = rn;
+        int n = rd ? node_data[rn].next : node_data[rn].prev;
+
+        while (n != rn) {
+
+          Node node = order_list[n];
+
+          if (embed_arc[node] != INVALID) {
+
+            // Merging components on the critical path
+            while (!merge_stack.empty()) {
+
+              // Component root
+              int cn = merge_stack.back().first;
+              bool cd = merge_stack.back().second;
+              merge_stack.pop_back();
+
+              // Parent of component
+              int dn = merge_stack.back().first;
+              bool dd = merge_stack.back().second;
+              merge_stack.pop_back();
+
+              Node parent = order_list[dn];
+
+              // Erasing from merge_roots
+              merge_roots[parent].pop_front();
+
+              Node child = order_list[cn - order_list.size()];
+
+              // Erasing from child_lists
+              if (child_lists[child].prev != INVALID) {
+                child_lists[child_lists[child].prev].next =
+                  child_lists[child].next;
+              } else {
+                child_lists[parent].first = child_lists[child].next;
+              }
+
+              if (child_lists[child].next != INVALID) {
+                child_lists[child_lists[child].next].prev =
+                  child_lists[child].prev;
+              }
+
+              // Merging arcs + flipping
+              Arc de = node_data[dn].first;
+              Arc ce = node_data[cn].first;
+
+              flip_map[order_list[cn - order_list.size()]] = cd != dd;
+              if (cd != dd) {
+                std::swap(arc_lists[ce].prev, arc_lists[ce].next);
+                ce = arc_lists[ce].prev;
+                std::swap(arc_lists[ce].prev, arc_lists[ce].next);
+              }
+
+              {
+                Arc dne = arc_lists[de].next;
+                Arc cne = arc_lists[ce].next;
+
+                arc_lists[de].next = cne;
+                arc_lists[ce].next = dne;
+
+                arc_lists[dne].prev = ce;
+                arc_lists[cne].prev = de;
+              }
+
+              if (dd) {
+                node_data[dn].first = ce;
+              }
+
+              // Merging external faces
+              {
+                int en = cn;
+                cn = cd ? node_data[cn].prev : node_data[cn].next;
+                cd = node_data[cn].next == en;
+
+                 if (node_data[cn].prev == node_data[cn].next &&
+                    node_data[cn].inverted) {
+                   cd = !cd;
+                 }
+              }
+
+              if (cd) node_data[cn].next = dn; else node_data[cn].prev = dn;
+              if (dd) node_data[dn].prev = cn; else node_data[dn].next = cn;
+
+            }
+
+            bool d = pn == node_data[n].prev;
+
+            if (node_data[n].prev == node_data[n].next &&
+                node_data[n].inverted) {
+              d = !d;
+            }
+
+            // Add new arc
+            {
+              Arc arc = embed_arc[node];
+              Arc re = node_data[rn].first;
+
+              arc_lists[arc_lists[re].next].prev = arc;
+              arc_lists[arc].next = arc_lists[re].next;
+              arc_lists[arc].prev = re;
+              arc_lists[re].next = arc;
+
+              if (!rd) {
+                node_data[rn].first = arc;
+              }
+
+              Arc rev = _graph.oppositeArc(arc);
+              Arc e = node_data[n].first;
+
+              arc_lists[arc_lists[e].next].prev = rev;
+              arc_lists[rev].next = arc_lists[e].next;
+              arc_lists[rev].prev = e;
+              arc_lists[e].next = rev;
+
+              if (d) {
+                node_data[n].first = rev;
+              }
+
+            }
+
+            // Embedding arc into external face
+            if (rd) node_data[rn].next = n; else node_data[rn].prev = n;
+            if (d) node_data[n].prev = rn; else node_data[n].next = rn;
+            pn = rn;
+
+            embed_arc[order_list[n]] = INVALID;
+          }
+
+          if (!merge_roots[node].empty()) {
+
+            bool d = pn == node_data[n].prev;
+            if (node_data[n].prev == node_data[n].next &&
+                node_data[n].inverted) {
+              d = !d;
+            }
+
+            merge_stack.push_back(std::make_pair(n, d));
+
+            int rn = merge_roots[node].front();
+
+            int xn = node_data[rn].next;
+            Node xnode = order_list[xn];
+
+            int yn = node_data[rn].prev;
+            Node ynode = order_list[yn];
+
+            bool rd;
+            if (!external(xnode, rorder, child_lists, ancestor_map, low_map)) {
+              rd = true;
+            } else if (!external(ynode, rorder, child_lists,
+                                 ancestor_map, low_map)) {
+              rd = false;
+            } else if (pertinent(xnode, embed_arc, merge_roots)) {
+              rd = true;
+            } else {
+              rd = false;
+            }
+
+            merge_stack.push_back(std::make_pair(rn, rd));
+
+            pn = rn;
+            n = rd ? xn : yn;
+
+          } else if (!external(node, rorder, child_lists,
+                               ancestor_map, low_map)) {
+            int nn = (node_data[n].next != pn ?
+                      node_data[n].next : node_data[n].prev);
+
+            bool nd = n == node_data[nn].prev;
+
+            if (nd) node_data[nn].prev = pn;
+            else node_data[nn].next = pn;
+
+            if (n == node_data[pn].prev) node_data[pn].prev = nn;
+            else node_data[pn].next = nn;
+
+            node_data[nn].inverted =
+              (node_data[nn].prev == node_data[nn].next && nd != rd);
+
+            n = nn;
+          }
+          else break;
+
+        }
+
+        if (!merge_stack.empty() || n == rn) {
+          break;
+        }
+      }
+    }
+
+    void initFace(const Node& node, ArcLists& arc_lists,
+                  NodeData& node_data, const PredMap& pred_map,
+                  const OrderMap& order_map, const OrderList& order_list) {
+      int n = order_map[node];
+      int rn = n + order_list.size();
+
+      node_data[n].next = node_data[n].prev = rn;
+      node_data[rn].next = node_data[rn].prev = n;
+
+      node_data[n].visited = order_list.size();
+      node_data[rn].visited = order_list.size();
+
+      node_data[n].inverted = false;
+      node_data[rn].inverted = false;
+
+      Arc arc = pred_map[node];
+      Arc rev = _graph.oppositeArc(arc);
+
+      node_data[rn].first = arc;
+      node_data[n].first = rev;
+
+      arc_lists[arc].prev = arc;
+      arc_lists[arc].next = arc;
+
+      arc_lists[rev].prev = rev;
+      arc_lists[rev].next = rev;
+
+    }
+
+    void mergeRemainingFaces(const Node& node, NodeData& node_data,
+                             OrderList& order_list, OrderMap& order_map,
+                             ChildLists& child_lists, ArcLists& arc_lists) {
+      while (child_lists[node].first != INVALID) {
+        int dd = order_map[node];
+        Node child = child_lists[node].first;
+        int cd = order_map[child] + order_list.size();
+        child_lists[node].first = child_lists[child].next;
+
+        Arc de = node_data[dd].first;
+        Arc ce = node_data[cd].first;
+
+        if (de != INVALID) {
+          Arc dne = arc_lists[de].next;
+          Arc cne = arc_lists[ce].next;
+
+          arc_lists[de].next = cne;
+          arc_lists[ce].next = dne;
+
+          arc_lists[dne].prev = ce;
+          arc_lists[cne].prev = de;
+        }
+
+        node_data[dd].first = ce;
+
+      }
+    }
+
+    void storeEmbedding(const Node& node, NodeData& node_data,
+                        OrderMap& order_map, PredMap& pred_map,
+                        ArcLists& arc_lists, FlipMap& flip_map) {
+
+      if (node_data[order_map[node]].first == INVALID) return;
+
+      if (pred_map[node] != INVALID) {
+        Node source = _graph.source(pred_map[node]);
+        flip_map[node] = flip_map[node] != flip_map[source];
+      }
+
+      Arc first = node_data[order_map[node]].first;
+      Arc prev = first;
+
+      Arc arc = flip_map[node] ?
+        arc_lists[prev].prev : arc_lists[prev].next;
+
+      _embedding[prev] = arc;
+
+      while (arc != first) {
+        Arc next = arc_lists[arc].prev == prev ?
+          arc_lists[arc].next : arc_lists[arc].prev;
+        prev = arc; arc = next;
+        _embedding[prev] = arc;
+      }
+    }
+
+
+    bool external(const Node& node, int rorder,
+                  ChildLists& child_lists, AncestorMap& ancestor_map,
+                  LowMap& low_map) {
+      Node child = child_lists[node].first;
+
+      if (child != INVALID) {
+        if (low_map[child] < rorder) return true;
+      }
+
+      if (ancestor_map[node] < rorder) return true;
+
+      return false;
+    }
+
+    bool pertinent(const Node& node, const EmbedArc& embed_arc,
+                   const MergeRoots& merge_roots) {
+      return !merge_roots[node].empty() || embed_arc[node] != INVALID;
+    }
+
+    int lowPoint(const Node& node, OrderMap& order_map, ChildLists& child_lists,
+                 AncestorMap& ancestor_map, LowMap& low_map) {
+      int low_point;
+
+      Node child = child_lists[node].first;
+
+      if (child != INVALID) {
+        low_point = low_map[child];
+      } else {
+        low_point = order_map[node];
+      }
+
+      if (low_point > ancestor_map[node]) {
+        low_point = ancestor_map[node];
+      }
+
+      return low_point;
+    }
+
+    int findComponentRoot(Node root, Node node, ChildLists& child_lists,
+                          OrderMap& order_map, OrderList& order_list) {
+
+      int order = order_map[root];
+      int norder = order_map[node];
+
+      Node child = child_lists[root].first;
+      while (child != INVALID) {
+        int corder = order_map[child];
+        if (corder > order && corder < norder) {
+          order = corder;
+        }
+        child = child_lists[child].next;
+      }
+      return order + order_list.size();
+    }
+
+    Node findPertinent(Node node, OrderMap& order_map, NodeData& node_data,
+                       EmbedArc& embed_arc, MergeRoots& merge_roots) {
+      Node wnode =_graph.target(node_data[order_map[node]].first);
+      while (!pertinent(wnode, embed_arc, merge_roots)) {
+        wnode = _graph.target(node_data[order_map[wnode]].first);
+      }
+      return wnode;
+    }
+
+
+    Node findExternal(Node node, int rorder, OrderMap& order_map,
+                      ChildLists& child_lists, AncestorMap& ancestor_map,
+                      LowMap& low_map, NodeData& node_data) {
+      Node wnode =_graph.target(node_data[order_map[node]].first);
+      while (!external(wnode, rorder, child_lists, ancestor_map, low_map)) {
+        wnode = _graph.target(node_data[order_map[wnode]].first);
+      }
+      return wnode;
+    }
+
+    void markCommonPath(Node node, int rorder, Node& wnode, Node& znode,
+                        OrderList& order_list, OrderMap& order_map,
+                        NodeData& node_data, ArcLists& arc_lists,
+                        EmbedArc& embed_arc, MergeRoots& merge_roots,
+                        ChildLists& child_lists, AncestorMap& ancestor_map,
+                        LowMap& low_map) {
+
+      Node cnode = node;
+      Node pred = INVALID;
+
+      while (true) {
+
+        bool pert = pertinent(cnode, embed_arc, merge_roots);
+        bool ext = external(cnode, rorder, child_lists, ancestor_map, low_map);
+
+        if (pert && ext) {
+          if (!merge_roots[cnode].empty()) {
+            int cn = merge_roots[cnode].back();
+
+            if (low_map[order_list[cn - order_list.size()]] < rorder) {
+              Arc arc = node_data[cn].first;
+              _kuratowski.set(arc, true);
+
+              pred = cnode;
+              cnode = _graph.target(arc);
+
+              continue;
+            }
+          }
+          wnode = znode = cnode;
+          return;
+
+        } else if (pert) {
+          wnode = cnode;
+
+          while (!external(cnode, rorder, child_lists, ancestor_map, low_map)) {
+            Arc arc = node_data[order_map[cnode]].first;
+
+            if (_graph.target(arc) == pred) {
+              arc = arc_lists[arc].next;
+            }
+            _kuratowski.set(arc, true);
+
+            Node next = _graph.target(arc);
+            pred = cnode; cnode = next;
+          }
+
+          znode = cnode;
+          return;
+
+        } else if (ext) {
+          znode = cnode;
+
+          while (!pertinent(cnode, embed_arc, merge_roots)) {
+            Arc arc = node_data[order_map[cnode]].first;
+
+            if (_graph.target(arc) == pred) {
+              arc = arc_lists[arc].next;
+            }
+            _kuratowski.set(arc, true);
+
+            Node next = _graph.target(arc);
+            pred = cnode; cnode = next;
+          }
+
+          wnode = cnode;
+          return;
+
+        } else {
+          Arc arc = node_data[order_map[cnode]].first;
+
+          if (_graph.target(arc) == pred) {
+            arc = arc_lists[arc].next;
+          }
+          _kuratowski.set(arc, true);
+
+          Node next = _graph.target(arc);
+          pred = cnode; cnode = next;
+        }
+
+      }
+
+    }
+
+    void orientComponent(Node root, int rn, OrderMap& order_map,
+                         PredMap& pred_map, NodeData& node_data,
+                         ArcLists& arc_lists, FlipMap& flip_map,
+                         TypeMap& type_map) {
+      node_data[order_map[root]].first = node_data[rn].first;
+      type_map[root] = 1;
+
+      std::vector<Node> st, qu;
+
+      st.push_back(root);
+      while (!st.empty()) {
+        Node node = st.back();
+        st.pop_back();
+        qu.push_back(node);
+
+        Arc arc = node_data[order_map[node]].first;
+
+        if (type_map[_graph.target(arc)] == 0) {
+          st.push_back(_graph.target(arc));
+          type_map[_graph.target(arc)] = 1;
+        }
+
+        Arc last = arc, pred = arc;
+        arc = arc_lists[arc].next;
+        while (arc != last) {
+
+          if (type_map[_graph.target(arc)] == 0) {
+            st.push_back(_graph.target(arc));
+            type_map[_graph.target(arc)] = 1;
+          }
+
+          Arc next = arc_lists[arc].next != pred ?
+            arc_lists[arc].next : arc_lists[arc].prev;
+          pred = arc; arc = next;
+        }
+
+      }
+
+      type_map[root] = 2;
+      flip_map[root] = false;
+
+      for (int i = 1; i < int(qu.size()); ++i) {
+
+        Node node = qu[i];
+
+        while (type_map[node] != 2) {
+          st.push_back(node);
+          type_map[node] = 2;
+          node = _graph.source(pred_map[node]);
+        }
+
+        bool flip = flip_map[node];
+
+        while (!st.empty()) {
+          node = st.back();
+          st.pop_back();
+
+          flip_map[node] = flip != flip_map[node];
+          flip = flip_map[node];
+
+          if (flip) {
+            Arc arc = node_data[order_map[node]].first;
+            std::swap(arc_lists[arc].prev, arc_lists[arc].next);
+            arc = arc_lists[arc].prev;
+            std::swap(arc_lists[arc].prev, arc_lists[arc].next);
+            node_data[order_map[node]].first = arc;
+          }
+        }
+      }
+
+      for (int i = 0; i < int(qu.size()); ++i) {
+
+        Arc arc = node_data[order_map[qu[i]]].first;
+        Arc last = arc, pred = arc;
+
+        arc = arc_lists[arc].next;
+        while (arc != last) {
+
+          if (arc_lists[arc].next == pred) {
+            std::swap(arc_lists[arc].next, arc_lists[arc].prev);
+          }
+          pred = arc; arc = arc_lists[arc].next;
+        }
+
+      }
+    }
+
+    void setFaceFlags(Node root, Node wnode, Node ynode, Node xnode,
+                      OrderMap& order_map, NodeData& node_data,
+                      TypeMap& type_map) {
+      Node node = _graph.target(node_data[order_map[root]].first);
+
+      while (node != ynode) {
+        type_map[node] = HIGHY;
+        node = _graph.target(node_data[order_map[node]].first);
+      }
+
+      while (node != wnode) {
+        type_map[node] = LOWY;
+        node = _graph.target(node_data[order_map[node]].first);
+      }
+
+      node = _graph.target(node_data[order_map[wnode]].first);
+
+      while (node != xnode) {
+        type_map[node] = LOWX;
+        node = _graph.target(node_data[order_map[node]].first);
+      }
+      type_map[node] = LOWX;
+
+      node = _graph.target(node_data[order_map[xnode]].first);
+      while (node != root) {
+        type_map[node] = HIGHX;
+        node = _graph.target(node_data[order_map[node]].first);
+      }
+
+      type_map[wnode] = PERTINENT;
+      type_map[root] = ROOT;
+    }
+
+    void findInternalPath(std::vector<Arc>& ipath,
+                          Node wnode, Node root, TypeMap& type_map,
+                          OrderMap& order_map, NodeData& node_data,
+                          ArcLists& arc_lists) {
+      std::vector<Arc> st;
+
+      Node node = wnode;
+
+      while (node != root) {
+        Arc arc = arc_lists[node_data[order_map[node]].first].next;
+        st.push_back(arc);
+        node = _graph.target(arc);
+      }
+
+      while (true) {
+        Arc arc = st.back();
+        if (type_map[_graph.target(arc)] == LOWX ||
+            type_map[_graph.target(arc)] == HIGHX) {
+          break;
+        }
+        if (type_map[_graph.target(arc)] == 2) {
+          type_map[_graph.target(arc)] = 3;
+
+          arc = arc_lists[_graph.oppositeArc(arc)].next;
+          st.push_back(arc);
+        } else {
+          st.pop_back();
+          arc = arc_lists[arc].next;
+
+          while (_graph.oppositeArc(arc) == st.back()) {
+            arc = st.back();
+            st.pop_back();
+            arc = arc_lists[arc].next;
+          }
+          st.push_back(arc);
+        }
+      }
+
+      for (int i = 0; i < int(st.size()); ++i) {
+        if (type_map[_graph.target(st[i])] != LOWY &&
+            type_map[_graph.target(st[i])] != HIGHY) {
+          for (; i < int(st.size()); ++i) {
+            ipath.push_back(st[i]);
+          }
+        }
+      }
+    }
+
+    void setInternalFlags(std::vector<Arc>& ipath, TypeMap& type_map) {
+      for (int i = 1; i < int(ipath.size()); ++i) {
+        type_map[_graph.source(ipath[i])] = INTERNAL;
+      }
+    }
+
+    void findPilePath(std::vector<Arc>& ppath,
+                      Node root, TypeMap& type_map, OrderMap& order_map,
+                      NodeData& node_data, ArcLists& arc_lists) {
+      std::vector<Arc> st;
+
+      st.push_back(_graph.oppositeArc(node_data[order_map[root]].first));
+      st.push_back(node_data[order_map[root]].first);
+
+      while (st.size() > 1) {
+        Arc arc = st.back();
+        if (type_map[_graph.target(arc)] == INTERNAL) {
+          break;
+        }
+        if (type_map[_graph.target(arc)] == 3) {
+          type_map[_graph.target(arc)] = 4;
+
+          arc = arc_lists[_graph.oppositeArc(arc)].next;
+          st.push_back(arc);
+        } else {
+          st.pop_back();
+          arc = arc_lists[arc].next;
+
+          while (!st.empty() && _graph.oppositeArc(arc) == st.back()) {
+            arc = st.back();
+            st.pop_back();
+            arc = arc_lists[arc].next;
+          }
+          st.push_back(arc);
+        }
+      }
+
+      for (int i = 1; i < int(st.size()); ++i) {
+        ppath.push_back(st[i]);
+      }
+    }
+
+
+    int markExternalPath(Node node, OrderMap& order_map,
+                         ChildLists& child_lists, PredMap& pred_map,
+                         AncestorMap& ancestor_map, LowMap& low_map) {
+      int lp = lowPoint(node, order_map, child_lists,
+                        ancestor_map, low_map);
+
+      if (ancestor_map[node] != lp) {
+        node = child_lists[node].first;
+        _kuratowski[pred_map[node]] = true;
+
+        while (ancestor_map[node] != lp) {
+          for (OutArcIt e(_graph, node); e != INVALID; ++e) {
+            Node tnode = _graph.target(e);
+            if (order_map[tnode] > order_map[node] && low_map[tnode] == lp) {
+              node = tnode;
+              _kuratowski[e] = true;
+              break;
+            }
+          }
+        }
+      }
+
+      for (OutArcIt e(_graph, node); e != INVALID; ++e) {
+        if (order_map[_graph.target(e)] == lp) {
+          _kuratowski[e] = true;
+          break;
+        }
+      }
+
+      return lp;
+    }
+
+    void markPertinentPath(Node node, OrderMap& order_map,
+                           NodeData& node_data, ArcLists& arc_lists,
+                           EmbedArc& embed_arc, MergeRoots& merge_roots) {
+      while (embed_arc[node] == INVALID) {
+        int n = merge_roots[node].front();
+        Arc arc = node_data[n].first;
+
+        _kuratowski.set(arc, true);
+
+        Node pred = node;
+        node = _graph.target(arc);
+        while (!pertinent(node, embed_arc, merge_roots)) {
+          arc = node_data[order_map[node]].first;
+          if (_graph.target(arc) == pred) {
+            arc = arc_lists[arc].next;
+          }
+          _kuratowski.set(arc, true);
+          pred = node;
+          node = _graph.target(arc);
+        }
+      }
+      _kuratowski.set(embed_arc[node], true);
+    }
+
+    void markPredPath(Node node, Node snode, PredMap& pred_map) {
+      while (node != snode) {
+        _kuratowski.set(pred_map[node], true);
+        node = _graph.source(pred_map[node]);
+      }
+    }
+
+    void markFacePath(Node ynode, Node xnode,
+                      OrderMap& order_map, NodeData& node_data) {
+      Arc arc = node_data[order_map[ynode]].first;
+      Node node = _graph.target(arc);
+      _kuratowski.set(arc, true);
+
+      while (node != xnode) {
+        arc = node_data[order_map[node]].first;
+        _kuratowski.set(arc, true);
+        node = _graph.target(arc);
+      }
+    }
+
+    void markInternalPath(std::vector<Arc>& path) {
+      for (int i = 0; i < int(path.size()); ++i) {
+        _kuratowski.set(path[i], true);
+      }
+    }
+
+    void markPilePath(std::vector<Arc>& path) {
+      for (int i = 0; i < int(path.size()); ++i) {
+        _kuratowski.set(path[i], true);
+      }
+    }
+
+    void isolateKuratowski(Arc arc, NodeData& node_data,
+                           ArcLists& arc_lists, FlipMap& flip_map,
+                           OrderMap& order_map, OrderList& order_list,
+                           PredMap& pred_map, ChildLists& child_lists,
+                           AncestorMap& ancestor_map, LowMap& low_map,
+                           EmbedArc& embed_arc, MergeRoots& merge_roots) {
+
+      Node root = _graph.source(arc);
+      Node enode = _graph.target(arc);
+
+      int rorder = order_map[root];
+
+      TypeMap type_map(_graph, 0);
+
+      int rn = findComponentRoot(root, enode, child_lists,
+                                 order_map, order_list);
+
+      Node xnode = order_list[node_data[rn].next];
+      Node ynode = order_list[node_data[rn].prev];
+
+      // Minor-A
+      {
+        while (!merge_roots[xnode].empty() || !merge_roots[ynode].empty()) {
+
+          if (!merge_roots[xnode].empty()) {
+            root = xnode;
+            rn = merge_roots[xnode].front();
+          } else {
+            root = ynode;
+            rn = merge_roots[ynode].front();
+          }
+
+          xnode = order_list[node_data[rn].next];
+          ynode = order_list[node_data[rn].prev];
+        }
+
+        if (root != _graph.source(arc)) {
+          orientComponent(root, rn, order_map, pred_map,
+                          node_data, arc_lists, flip_map, type_map);
+          markFacePath(root, root, order_map, node_data);
+          int xlp = markExternalPath(xnode, order_map, child_lists,
+                                     pred_map, ancestor_map, low_map);
+          int ylp = markExternalPath(ynode, order_map, child_lists,
+                                     pred_map, ancestor_map, low_map);
+          markPredPath(root, order_list[xlp < ylp ? xlp : ylp], pred_map);
+          Node lwnode = findPertinent(ynode, order_map, node_data,
+                                      embed_arc, merge_roots);
+
+          markPertinentPath(lwnode, order_map, node_data, arc_lists,
+                            embed_arc, merge_roots);
+
+          return;
+        }
+      }
+
+      orientComponent(root, rn, order_map, pred_map,
+                      node_data, arc_lists, flip_map, type_map);
+
+      Node wnode = findPertinent(ynode, order_map, node_data,
+                                 embed_arc, merge_roots);
+      setFaceFlags(root, wnode, ynode, xnode, order_map, node_data, type_map);
+
+
+      //Minor-B
+      if (!merge_roots[wnode].empty()) {
+        int cn = merge_roots[wnode].back();
+        Node rep = order_list[cn - order_list.size()];
+        if (low_map[rep] < rorder) {
+          markFacePath(root, root, order_map, node_data);
+          int xlp = markExternalPath(xnode, order_map, child_lists,
+                                     pred_map, ancestor_map, low_map);
+          int ylp = markExternalPath(ynode, order_map, child_lists,
+                                     pred_map, ancestor_map, low_map);
+
+          Node lwnode, lznode;
+          markCommonPath(wnode, rorder, lwnode, lznode, order_list,
+                         order_map, node_data, arc_lists, embed_arc,
+                         merge_roots, child_lists, ancestor_map, low_map);
+
+          markPertinentPath(lwnode, order_map, node_data, arc_lists,
+                            embed_arc, merge_roots);
+          int zlp = markExternalPath(lznode, order_map, child_lists,
+                                     pred_map, ancestor_map, low_map);
+
+          int minlp = xlp < ylp ? xlp : ylp;
+          if (zlp < minlp) minlp = zlp;
+
+          int maxlp = xlp > ylp ? xlp : ylp;
+          if (zlp > maxlp) maxlp = zlp;
+
+          markPredPath(order_list[maxlp], order_list[minlp], pred_map);
+
+          return;
+        }
+      }
+
+      Node pxnode, pynode;
+      std::vector<Arc> ipath;
+      findInternalPath(ipath, wnode, root, type_map, order_map,
+                       node_data, arc_lists);
+      setInternalFlags(ipath, type_map);
+      pynode = _graph.source(ipath.front());
+      pxnode = _graph.target(ipath.back());
+
+      wnode = findPertinent(pynode, order_map, node_data,
+                            embed_arc, merge_roots);
+
+      // Minor-C
+      {
+        if (type_map[_graph.source(ipath.front())] == HIGHY) {
+          if (type_map[_graph.target(ipath.back())] == HIGHX) {
+            markFacePath(xnode, pxnode, order_map, node_data);
+          }
+          markFacePath(root, xnode, order_map, node_data);
+          markPertinentPath(wnode, order_map, node_data, arc_lists,
+                            embed_arc, merge_roots);
+          markInternalPath(ipath);
+          int xlp = markExternalPath(xnode, order_map, child_lists,
+                                     pred_map, ancestor_map, low_map);
+          int ylp = markExternalPath(ynode, order_map, child_lists,
+                                     pred_map, ancestor_map, low_map);
+          markPredPath(root, order_list[xlp < ylp ? xlp : ylp], pred_map);
+          return;
+        }
+
+        if (type_map[_graph.target(ipath.back())] == HIGHX) {
+          markFacePath(ynode, root, order_map, node_data);
+          markPertinentPath(wnode, order_map, node_data, arc_lists,
+                            embed_arc, merge_roots);
+          markInternalPath(ipath);
+          int xlp = markExternalPath(xnode, order_map, child_lists,
+                                     pred_map, ancestor_map, low_map);
+          int ylp = markExternalPath(ynode, order_map, child_lists,
+                                     pred_map, ancestor_map, low_map);
+          markPredPath(root, order_list[xlp < ylp ? xlp : ylp], pred_map);
+          return;
+        }
+      }
+
+      std::vector<Arc> ppath;
+      findPilePath(ppath, root, type_map, order_map, node_data, arc_lists);
+
+      // Minor-D
+      if (!ppath.empty()) {
+        markFacePath(ynode, xnode, order_map, node_data);
+        markPertinentPath(wnode, order_map, node_data, arc_lists,
+                          embed_arc, merge_roots);
+        markPilePath(ppath);
+        markInternalPath(ipath);
+        int xlp = markExternalPath(xnode, order_map, child_lists,
+                                   pred_map, ancestor_map, low_map);
+        int ylp = markExternalPath(ynode, order_map, child_lists,
+                                   pred_map, ancestor_map, low_map);
+        markPredPath(root, order_list[xlp < ylp ? xlp : ylp], pred_map);
+        return;
+      }
+
+      // Minor-E*
+      {
+
+        if (!external(wnode, rorder, child_lists, ancestor_map, low_map)) {
+          Node znode = findExternal(pynode, rorder, order_map,
+                                    child_lists, ancestor_map,
+                                    low_map, node_data);
+
+          if (type_map[znode] == LOWY) {
+            markFacePath(root, xnode, order_map, node_data);
+            markPertinentPath(wnode, order_map, node_data, arc_lists,
+                              embed_arc, merge_roots);
+            markInternalPath(ipath);
+            int xlp = markExternalPath(xnode, order_map, child_lists,
+                                       pred_map, ancestor_map, low_map);
+            int zlp = markExternalPath(znode, order_map, child_lists,
+                                       pred_map, ancestor_map, low_map);
+            markPredPath(root, order_list[xlp < zlp ? xlp : zlp], pred_map);
+          } else {
+            markFacePath(ynode, root, order_map, node_data);
+            markPertinentPath(wnode, order_map, node_data, arc_lists,
+                              embed_arc, merge_roots);
+            markInternalPath(ipath);
+            int ylp = markExternalPath(ynode, order_map, child_lists,
+                                       pred_map, ancestor_map, low_map);
+            int zlp = markExternalPath(znode, order_map, child_lists,
+                                       pred_map, ancestor_map, low_map);
+            markPredPath(root, order_list[ylp < zlp ? ylp : zlp], pred_map);
+          }
+          return;
+        }
+
+        int xlp = markExternalPath(xnode, order_map, child_lists,
+                                   pred_map, ancestor_map, low_map);
+        int ylp = markExternalPath(ynode, order_map, child_lists,
+                                   pred_map, ancestor_map, low_map);
+        int wlp = markExternalPath(wnode, order_map, child_lists,
+                                   pred_map, ancestor_map, low_map);
+
+        if (wlp > xlp && wlp > ylp) {
+          markFacePath(root, root, order_map, node_data);
+          markPredPath(root, order_list[xlp < ylp ? xlp : ylp], pred_map);
+          return;
+        }
+
+        markInternalPath(ipath);
+        markPertinentPath(wnode, order_map, node_data, arc_lists,
+                          embed_arc, merge_roots);
+
+        if (xlp > ylp && xlp > wlp) {
+          markFacePath(root, pynode, order_map, node_data);
+          markFacePath(wnode, xnode, order_map, node_data);
+          markPredPath(root, order_list[ylp < wlp ? ylp : wlp], pred_map);
+          return;
+        }
+
+        if (ylp > xlp && ylp > wlp) {
+          markFacePath(pxnode, root, order_map, node_data);
+          markFacePath(ynode, wnode, order_map, node_data);
+          markPredPath(root, order_list[xlp < wlp ? xlp : wlp], pred_map);
+          return;
+        }
+
+        if (pynode != ynode) {
+          markFacePath(pxnode, wnode, order_map, node_data);
+
+          int minlp = xlp < ylp ? xlp : ylp;
+          if (wlp < minlp) minlp = wlp;
+
+          int maxlp = xlp > ylp ? xlp : ylp;
+          if (wlp > maxlp) maxlp = wlp;
+
+          markPredPath(order_list[maxlp], order_list[minlp], pred_map);
+          return;
+        }
+
+        if (pxnode != xnode) {
+          markFacePath(wnode, pynode, order_map, node_data);
+
+          int minlp = xlp < ylp ? xlp : ylp;
+          if (wlp < minlp) minlp = wlp;
+
+          int maxlp = xlp > ylp ? xlp : ylp;
+          if (wlp > maxlp) maxlp = wlp;
+
+          markPredPath(order_list[maxlp], order_list[minlp], pred_map);
+          return;
+        }
+
+        markFacePath(root, root, order_map, node_data);
+        int minlp = xlp < ylp ? xlp : ylp;
+        if (wlp < minlp) minlp = wlp;
+        markPredPath(root, order_list[minlp], pred_map);
+        return;
+      }
+
+    }
+
+  };
+
+  namespace _planarity_bits {
+
+    template <typename Graph, typename EmbeddingMap>
+    void makeConnected(Graph& graph, EmbeddingMap& embedding) {
+      DfsVisitor<Graph> null_visitor;
+      DfsVisit<Graph, DfsVisitor<Graph> > dfs(graph, null_visitor);
+      dfs.init();
+
+      typename Graph::Node u = INVALID;
+      for (typename Graph::NodeIt n(graph); n != INVALID; ++n) {
+        if (!dfs.reached(n)) {
+          dfs.addSource(n);
+          dfs.start();
+          if (u == INVALID) {
+            u = n;
+          } else {
+            typename Graph::Node v = n;
+
+            typename Graph::Arc ue = typename Graph::OutArcIt(graph, u);
+            typename Graph::Arc ve = typename Graph::OutArcIt(graph, v);
+
+            typename Graph::Arc e = graph.direct(graph.addEdge(u, v), true);
+
+            if (ue != INVALID) {
+              embedding[e] = embedding[ue];
+              embedding[ue] = e;
+            } else {
+              embedding[e] = e;
+            }
+
+            if (ve != INVALID) {
+              embedding[graph.oppositeArc(e)] = embedding[ve];
+              embedding[ve] = graph.oppositeArc(e);
+            } else {
+              embedding[graph.oppositeArc(e)] = graph.oppositeArc(e);
+            }
+          }
+        }
+      }
+    }
+
+    template <typename Graph, typename EmbeddingMap>
+    void makeBiNodeConnected(Graph& graph, EmbeddingMap& embedding) {
+      typename Graph::template ArcMap<bool> processed(graph);
+
+      std::vector<typename Graph::Arc> arcs;
+      for (typename Graph::ArcIt e(graph); e != INVALID; ++e) {
+        arcs.push_back(e);
+      }
+
+      IterableBoolMap<Graph, typename Graph::Node> visited(graph, false);
+
+      for (int i = 0; i < int(arcs.size()); ++i) {
+        typename Graph::Arc pp = arcs[i];
+        if (processed[pp]) continue;
+
+        typename Graph::Arc e = embedding[graph.oppositeArc(pp)];
+        processed[e] = true;
+        visited.set(graph.source(e), true);
+
+        typename Graph::Arc p = e, l = e;
+        e = embedding[graph.oppositeArc(e)];
+
+        while (e != l) {
+          processed[e] = true;
+
+          if (visited[graph.source(e)]) {
+
+            typename Graph::Arc n =
+              graph.direct(graph.addEdge(graph.source(p),
+                                           graph.target(e)), true);
+            embedding[n] = p;
+            embedding[graph.oppositeArc(pp)] = n;
+
+            embedding[graph.oppositeArc(n)] =
+              embedding[graph.oppositeArc(e)];
+            embedding[graph.oppositeArc(e)] =
+              graph.oppositeArc(n);
+
+            p = n;
+            e = embedding[graph.oppositeArc(n)];
+          } else {
+            visited.set(graph.source(e), true);
+            pp = p;
+            p = e;
+            e = embedding[graph.oppositeArc(e)];
+          }
+        }
+        visited.setAll(false);
+      }
+    }
+
+
+    template <typename Graph, typename EmbeddingMap>
+    void makeMaxPlanar(Graph& graph, EmbeddingMap& embedding) {
+
+      typename Graph::template NodeMap<int> degree(graph);
+
+      for (typename Graph::NodeIt n(graph); n != INVALID; ++n) {
+        degree[n] = countIncEdges(graph, n);
+      }
+
+      typename Graph::template ArcMap<bool> processed(graph);
+      IterableBoolMap<Graph, typename Graph::Node> visited(graph, false);
+
+      std::vector<typename Graph::Arc> arcs;
+      for (typename Graph::ArcIt e(graph); e != INVALID; ++e) {
+        arcs.push_back(e);
+      }
+
+      for (int i = 0; i < int(arcs.size()); ++i) {
+        typename Graph::Arc e = arcs[i];
+
+        if (processed[e]) continue;
+        processed[e] = true;
+
+        typename Graph::Arc mine = e;
+        int mind = degree[graph.source(e)];
+
+        int face_size = 1;
+
+        typename Graph::Arc l = e;
+        e = embedding[graph.oppositeArc(e)];
+        while (l != e) {
+          processed[e] = true;
+
+          ++face_size;
+
+          if (degree[graph.source(e)] < mind) {
+            mine = e;
+            mind = degree[graph.source(e)];
+          }
+
+          e = embedding[graph.oppositeArc(e)];
+        }
+
+        if (face_size < 4) {
+          continue;
+        }
+
+        typename Graph::Node s = graph.source(mine);
+        for (typename Graph::OutArcIt e(graph, s); e != INVALID; ++e) {
+          visited.set(graph.target(e), true);
+        }
+
+        typename Graph::Arc oppe = INVALID;
+
+        e = embedding[graph.oppositeArc(mine)];
+        e = embedding[graph.oppositeArc(e)];
+        while (graph.target(e) != s) {
+          if (visited[graph.source(e)]) {
+            oppe = e;
+            break;
+          }
+          e = embedding[graph.oppositeArc(e)];
+        }
+        visited.setAll(false);
+
+        if (oppe == INVALID) {
+
+          e = embedding[graph.oppositeArc(mine)];
+          typename Graph::Arc pn = mine, p = e;
+
+          e = embedding[graph.oppositeArc(e)];
+          while (graph.target(e) != s) {
+            typename Graph::Arc n =
+              graph.direct(graph.addEdge(s, graph.source(e)), true);
+
+            embedding[n] = pn;
+            embedding[graph.oppositeArc(n)] = e;
+            embedding[graph.oppositeArc(p)] = graph.oppositeArc(n);
+
+            pn = n;
+
+            p = e;
+            e = embedding[graph.oppositeArc(e)];
+          }
+
+          embedding[graph.oppositeArc(e)] = pn;
+
+        } else {
+
+          mine = embedding[graph.oppositeArc(mine)];
+          s = graph.source(mine);
+          oppe = embedding[graph.oppositeArc(oppe)];
+          typename Graph::Node t = graph.source(oppe);
+
+          typename Graph::Arc ce = graph.direct(graph.addEdge(s, t), true);
+          embedding[ce] = mine;
+          embedding[graph.oppositeArc(ce)] = oppe;
+
+          typename Graph::Arc pn = ce, p = oppe;
+          e = embedding[graph.oppositeArc(oppe)];
+          while (graph.target(e) != s) {
+            typename Graph::Arc n =
+              graph.direct(graph.addEdge(s, graph.source(e)), true);
+
+            embedding[n] = pn;
+            embedding[graph.oppositeArc(n)] = e;
+            embedding[graph.oppositeArc(p)] = graph.oppositeArc(n);
+
+            pn = n;
+
+            p = e;
+            e = embedding[graph.oppositeArc(e)];
+
+          }
+          embedding[graph.oppositeArc(e)] = pn;
+
+          pn = graph.oppositeArc(ce), p = mine;
+          e = embedding[graph.oppositeArc(mine)];
+          while (graph.target(e) != t) {
+            typename Graph::Arc n =
+              graph.direct(graph.addEdge(t, graph.source(e)), true);
+
+            embedding[n] = pn;
+            embedding[graph.oppositeArc(n)] = e;
+            embedding[graph.oppositeArc(p)] = graph.oppositeArc(n);
+
+            pn = n;
+
+            p = e;
+            e = embedding[graph.oppositeArc(e)];
+
+          }
+          embedding[graph.oppositeArc(e)] = pn;
+        }
+      }
+    }
+
+  }
+
+  /// \ingroup planar
+  ///
+  /// \brief Schnyder's planar drawing algorithm
+  ///
+  /// The planar drawing algorithm calculates positions for the nodes
+  /// in the plane which coordinates satisfy that if the arcs are
+  /// represented with straight lines then they will not intersect
+  /// each other.
+  ///
+  /// Scnyder's algorithm embeds the graph on \c (n-2,n-2) size grid,
+  /// i.e. each node will be located in the \c [0,n-2]x[0,n-2] square.
+  /// The time complexity of the algorithm is O(n).
+  template <typename Graph>
+  class PlanarDrawing {
+  public:
+
+    TEMPLATE_GRAPH_TYPEDEFS(Graph);
+
+    /// \brief The point type for store coordinates
+    typedef dim2::Point<int> Point;
+    /// \brief The map type for store coordinates
+    typedef typename Graph::template NodeMap<Point> PointMap;
+
+
+    /// \brief Constructor
+    ///
+    /// Constructor
+    /// \pre The graph should be simple, i.e. loop and parallel arc free.
+    PlanarDrawing(const Graph& graph)
+      : _graph(graph), _point_map(graph) {}
+
+  private:
+
+    template <typename AuxGraph, typename AuxEmbeddingMap>
+    void drawing(const AuxGraph& graph,
+                 const AuxEmbeddingMap& next,
+                 PointMap& point_map) {
+      TEMPLATE_GRAPH_TYPEDEFS(AuxGraph);
+
+      typename AuxGraph::template ArcMap<Arc> prev(graph);
+
+      for (NodeIt n(graph); n != INVALID; ++n) {
+        Arc e = OutArcIt(graph, n);
+
+        Arc p = e, l = e;
+
+        e = next[e];
+        while (e != l) {
+          prev[e] = p;
+          p = e;
+          e = next[e];
+        }
+        prev[e] = p;
+      }
+
+      Node anode, bnode, cnode;
+
+      {
+        Arc e = ArcIt(graph);
+        anode = graph.source(e);
+        bnode = graph.target(e);
+        cnode = graph.target(next[graph.oppositeArc(e)]);
+      }
+
+      IterableBoolMap<AuxGraph, Node> proper(graph, false);
+      typename AuxGraph::template NodeMap<int> conn(graph, -1);
+
+      conn[anode] = conn[bnode] = -2;
+      {
+        for (OutArcIt e(graph, anode); e != INVALID; ++e) {
+          Node m = graph.target(e);
+          if (conn[m] == -1) {
+            conn[m] = 1;
+          }
+        }
+        conn[cnode] = 2;
+
+        for (OutArcIt e(graph, bnode); e != INVALID; ++e) {
+          Node m = graph.target(e);
+          if (conn[m] == -1) {
+            conn[m] = 1;
+          } else if (conn[m] != -2) {
+            conn[m] += 1;
+            Arc pe = graph.oppositeArc(e);
+            if (conn[graph.target(next[pe])] == -2) {
+              conn[m] -= 1;
+            }
+            if (conn[graph.target(prev[pe])] == -2) {
+              conn[m] -= 1;
+            }
+
+            proper.set(m, conn[m] == 1);
+          }
+        }
+      }
+
+
+      typename AuxGraph::template ArcMap<int> angle(graph, -1);
+
+      while (proper.trueNum() != 0) {
+        Node n = typename IterableBoolMap<AuxGraph, Node>::TrueIt(proper);
+        proper.set(n, false);
+        conn[n] = -2;
+
+        for (OutArcIt e(graph, n); e != INVALID; ++e) {
+          Node m = graph.target(e);
+          if (conn[m] == -1) {
+            conn[m] = 1;
+          } else if (conn[m] != -2) {
+            conn[m] += 1;
+            Arc pe = graph.oppositeArc(e);
+            if (conn[graph.target(next[pe])] == -2) {
+              conn[m] -= 1;
+            }
+            if (conn[graph.target(prev[pe])] == -2) {
+              conn[m] -= 1;
+            }
+
+            proper.set(m, conn[m] == 1);
+          }
+        }
+
+        {
+          Arc e = OutArcIt(graph, n);
+          Arc p = e, l = e;
+
+          e = next[e];
+          while (e != l) {
+
+            if (conn[graph.target(e)] == -2 && conn[graph.target(p)] == -2) {
+              Arc f = e;
+              angle[f] = 0;
+              f = next[graph.oppositeArc(f)];
+              angle[f] = 1;
+              f = next[graph.oppositeArc(f)];
+              angle[f] = 2;
+            }
+
+            p = e;
+            e = next[e];
+          }
+
+          if (conn[graph.target(e)] == -2 && conn[graph.target(p)] == -2) {
+            Arc f = e;
+            angle[f] = 0;
+            f = next[graph.oppositeArc(f)];
+            angle[f] = 1;
+            f = next[graph.oppositeArc(f)];
+            angle[f] = 2;
+          }
+        }
+      }
+
+      typename AuxGraph::template NodeMap<Node> apred(graph, INVALID);
+      typename AuxGraph::template NodeMap<Node> bpred(graph, INVALID);
+      typename AuxGraph::template NodeMap<Node> cpred(graph, INVALID);
+
+      typename AuxGraph::template NodeMap<int> apredid(graph, -1);
+      typename AuxGraph::template NodeMap<int> bpredid(graph, -1);
+      typename AuxGraph::template NodeMap<int> cpredid(graph, -1);
+
+      for (ArcIt e(graph); e != INVALID; ++e) {
+        if (angle[e] == angle[next[e]]) {
+          switch (angle[e]) {
+          case 2:
+            apred[graph.target(e)] = graph.source(e);
+            apredid[graph.target(e)] = graph.id(graph.source(e));
+            break;
+          case 1:
+            bpred[graph.target(e)] = graph.source(e);
+            bpredid[graph.target(e)] = graph.id(graph.source(e));
+            break;
+          case 0:
+            cpred[graph.target(e)] = graph.source(e);
+            cpredid[graph.target(e)] = graph.id(graph.source(e));
+            break;
+          }
+        }
+      }
+
+      cpred[anode] = INVALID;
+      cpred[bnode] = INVALID;
+
+      std::vector<Node> aorder, border, corder;
+
+      {
+        typename AuxGraph::template NodeMap<bool> processed(graph, false);
+        std::vector<Node> st;
+        for (NodeIt n(graph); n != INVALID; ++n) {
+          if (!processed[n] && n != bnode && n != cnode) {
+            st.push_back(n);
+            processed[n] = true;
+            Node m = apred[n];
+            while (m != INVALID && !processed[m]) {
+              st.push_back(m);
+              processed[m] = true;
+              m = apred[m];
+            }
+            while (!st.empty()) {
+              aorder.push_back(st.back());
+              st.pop_back();
+            }
+          }
+        }
+      }
+
+      {
+        typename AuxGraph::template NodeMap<bool> processed(graph, false);
+        std::vector<Node> st;
+        for (NodeIt n(graph); n != INVALID; ++n) {
+          if (!processed[n] && n != cnode && n != anode) {
+            st.push_back(n);
+            processed[n] = true;
+            Node m = bpred[n];
+            while (m != INVALID && !processed[m]) {
+              st.push_back(m);
+              processed[m] = true;
+              m = bpred[m];
+            }
+            while (!st.empty()) {
+              border.push_back(st.back());
+              st.pop_back();
+            }
+          }
+        }
+      }
+
+      {
+        typename AuxGraph::template NodeMap<bool> processed(graph, false);
+        std::vector<Node> st;
+        for (NodeIt n(graph); n != INVALID; ++n) {
+          if (!processed[n] && n != anode && n != bnode) {
+            st.push_back(n);
+            processed[n] = true;
+            Node m = cpred[n];
+            while (m != INVALID && !processed[m]) {
+              st.push_back(m);
+              processed[m] = true;
+              m = cpred[m];
+            }
+            while (!st.empty()) {
+              corder.push_back(st.back());
+              st.pop_back();
+            }
+          }
+        }
+      }
+
+      typename AuxGraph::template NodeMap<int> atree(graph, 0);
+      for (int i = aorder.size() - 1; i >= 0; --i) {
+        Node n = aorder[i];
+        atree[n] = 1;
+        for (OutArcIt e(graph, n); e != INVALID; ++e) {
+          if (apred[graph.target(e)] == n) {
+            atree[n] += atree[graph.target(e)];
+          }
+        }
+      }
+
+      typename AuxGraph::template NodeMap<int> btree(graph, 0);
+      for (int i = border.size() - 1; i >= 0; --i) {
+        Node n = border[i];
+        btree[n] = 1;
+        for (OutArcIt e(graph, n); e != INVALID; ++e) {
+          if (bpred[graph.target(e)] == n) {
+            btree[n] += btree[graph.target(e)];
+          }
+        }
+      }
+
+      typename AuxGraph::template NodeMap<int> apath(graph, 0);
+      apath[bnode] = apath[cnode] = 1;
+      typename AuxGraph::template NodeMap<int> apath_btree(graph, 0);
+      apath_btree[bnode] = btree[bnode];
+      for (int i = 1; i < int(aorder.size()); ++i) {
+        Node n = aorder[i];
+        apath[n] = apath[apred[n]] + 1;
+        apath_btree[n] = btree[n] + apath_btree[apred[n]];
+      }
+
+      typename AuxGraph::template NodeMap<int> bpath_atree(graph, 0);
+      bpath_atree[anode] = atree[anode];
+      for (int i = 1; i < int(border.size()); ++i) {
+        Node n = border[i];
+        bpath_atree[n] = atree[n] + bpath_atree[bpred[n]];
+      }
+
+      typename AuxGraph::template NodeMap<int> cpath(graph, 0);
+      cpath[anode] = cpath[bnode] = 1;
+      typename AuxGraph::template NodeMap<int> cpath_atree(graph, 0);
+      cpath_atree[anode] = atree[anode];
+      typename AuxGraph::template NodeMap<int> cpath_btree(graph, 0);
+      cpath_btree[bnode] = btree[bnode];
+      for (int i = 1; i < int(corder.size()); ++i) {
+        Node n = corder[i];
+        cpath[n] = cpath[cpred[n]] + 1;
+        cpath_atree[n] = atree[n] + cpath_atree[cpred[n]];
+        cpath_btree[n] = btree[n] + cpath_btree[cpred[n]];
+      }
+
+      typename AuxGraph::template NodeMap<int> third(graph);
+      for (NodeIt n(graph); n != INVALID; ++n) {
+        point_map[n].x =
+          bpath_atree[n] + cpath_atree[n] - atree[n] - cpath[n] + 1;
+        point_map[n].y =
+          cpath_btree[n] + apath_btree[n] - btree[n] - apath[n] + 1;
+      }
+
+    }
+
+  public:
+
+    /// \brief Calculates the node positions
+    ///
+    /// This function calculates the node positions.
+    /// \return %True if the graph is planar.
+    bool run() {
+      PlanarEmbedding<Graph> pe(_graph);
+      if (!pe.run()) return false;
+
+      run(pe);
+      return true;
+    }
+
+    /// \brief Calculates the node positions according to a
+    /// combinatorical embedding
+    ///
+    /// This function calculates the node locations. The \c embedding
+    /// parameter should contain a valid combinatorical embedding, i.e.
+    /// a valid cyclic order of the arcs.
+    template <typename EmbeddingMap>
+    void run(const EmbeddingMap& embedding) {
+      typedef SmartEdgeSet<Graph> AuxGraph;
+
+      if (3 * countNodes(_graph) - 6 == countEdges(_graph)) {
+        drawing(_graph, embedding, _point_map);
+        return;
+      }
+
+      AuxGraph aux_graph(_graph);
+      typename AuxGraph::template ArcMap<typename AuxGraph::Arc>
+        aux_embedding(aux_graph);
+
+      {
+
+        typename Graph::template EdgeMap<typename AuxGraph::Edge>
+          ref(_graph);
+
+        for (EdgeIt e(_graph); e != INVALID; ++e) {
+          ref[e] = aux_graph.addEdge(_graph.u(e), _graph.v(e));
+        }
+
+        for (EdgeIt e(_graph); e != INVALID; ++e) {
+          Arc ee = embedding[_graph.direct(e, true)];
+          aux_embedding[aux_graph.direct(ref[e], true)] =
+            aux_graph.direct(ref[ee], _graph.direction(ee));
+          ee = embedding[_graph.direct(e, false)];
+          aux_embedding[aux_graph.direct(ref[e], false)] =
+            aux_graph.direct(ref[ee], _graph.direction(ee));
+        }
+      }
+      _planarity_bits::makeConnected(aux_graph, aux_embedding);
+      _planarity_bits::makeBiNodeConnected(aux_graph, aux_embedding);
+      _planarity_bits::makeMaxPlanar(aux_graph, aux_embedding);
+      drawing(aux_graph, aux_embedding, _point_map);
+    }
+
+    /// \brief The coordinate of the given node
+    ///
+    /// The coordinate of the given node.
+    Point operator[](const Node& node) const {
+      return _point_map[node];
+    }
+
+    /// \brief Returns the grid embedding in a \e NodeMap.
+    ///
+    /// Returns the grid embedding in a \e NodeMap of \c dim2::Point<int> .
+    const PointMap& coords() const {
+      return _point_map;
+    }
+
+  private:
+
+    const Graph& _graph;
+    PointMap _point_map;
+
+  };
+
+  namespace _planarity_bits {
+
+    template <typename ColorMap>
+    class KempeFilter {
+    public:
+      typedef typename ColorMap::Key Key;
+      typedef bool Value;
+
+      KempeFilter(const ColorMap& color_map,
+                  const typename ColorMap::Value& first,
+                  const typename ColorMap::Value& second)
+        : _color_map(color_map), _first(first), _second(second) {}
+
+      Value operator[](const Key& key) const {
+        return _color_map[key] == _first || _color_map[key] == _second;
+      }
+
+    private:
+      const ColorMap& _color_map;
+      typename ColorMap::Value _first, _second;
+    };
+  }
+
+  /// \ingroup planar
+  ///
+  /// \brief Coloring planar graphs
+  ///
+  /// The graph coloring problem is the coloring of the graph nodes
+  /// that there are not adjacent nodes with the same color. The
+  /// planar graphs can be always colored with four colors, it is
+  /// proved by Appel and Haken and their proofs provide a quadratic
+  /// time algorithm for four coloring, but it could not be used to
+  /// implement efficient algorithm. The five and six coloring can be
+  /// made in linear time, but in this class the five coloring has
+  /// quadratic worst case time complexity. The two coloring (if
+  /// possible) is solvable with a graph search algorithm and it is
+  /// implemented in \ref bipartitePartitions() function in LEMON. To
+  /// decide whether the planar graph is three colorable is
+  /// NP-complete.
+  ///
+  /// This class contains member functions for calculate colorings
+  /// with five and six colors. The six coloring algorithm is a simple
+  /// greedy coloring on the backward minimum outgoing order of nodes.
+  /// This order can be computed as in each phase the node with least
+  /// outgoing arcs to unprocessed nodes is chosen. This order
+  /// guarantees that when a node is chosen for coloring it has at
+  /// most five already colored adjacents. The five coloring algorithm
+  /// use the same method, but if the greedy approach fails to color
+  /// with five colors, i.e. the node has five already different
+  /// colored neighbours, it swaps the colors in one of the connected
+  /// two colored sets with the Kempe recoloring method.
+  template <typename Graph>
+  class PlanarColoring {
+  public:
+
+    TEMPLATE_GRAPH_TYPEDEFS(Graph);
+
+    /// \brief The map type for store color indexes
+    typedef typename Graph::template NodeMap<int> IndexMap;
+    /// \brief The map type for store colors
+    typedef ComposeMap<Palette, IndexMap> ColorMap;
+
+    /// \brief Constructor
+    ///
+    /// Constructor
+    /// \pre The graph should be simple, i.e. loop and parallel arc free.
+    PlanarColoring(const Graph& graph)
+      : _graph(graph), _color_map(graph), _palette(0) {
+      _palette.add(Color(1,0,0));
+      _palette.add(Color(0,1,0));
+      _palette.add(Color(0,0,1));
+      _palette.add(Color(1,1,0));
+      _palette.add(Color(1,0,1));
+      _palette.add(Color(0,1,1));
+    }
+
+    /// \brief Returns the \e NodeMap of color indexes
+    ///
+    /// Returns the \e NodeMap of color indexes. The values are in the
+    /// range \c [0..4] or \c [0..5] according to the coloring method.
+    IndexMap colorIndexMap() const {
+      return _color_map;
+    }
+
+    /// \brief Returns the \e NodeMap of colors
+    ///
+    /// Returns the \e NodeMap of colors. The values are five or six
+    /// distinct \ref lemon::Color "colors".
+    ColorMap colorMap() const {
+      return composeMap(_palette, _color_map);
+    }
+
+    /// \brief Returns the color index of the node
+    ///
+    /// Returns the color index of the node. The values are in the
+    /// range \c [0..4] or \c [0..5] according to the coloring method.
+    int colorIndex(const Node& node) const {
+      return _color_map[node];
+    }
+
+    /// \brief Returns the color of the node
+    ///
+    /// Returns the color of the node. The values are five or six
+    /// distinct \ref lemon::Color "colors".
+    Color color(const Node& node) const {
+      return _palette[_color_map[node]];
+    }
+
+
+    /// \brief Calculates a coloring with at most six colors
+    ///
+    /// This function calculates a coloring with at most six colors. The time
+    /// complexity of this variant is linear in the size of the graph.
+    /// \return %True when the algorithm could color the graph with six color.
+    /// If the algorithm fails, then the graph could not be planar.
+    /// \note This function can return true if the graph is not
+    /// planar but it can be colored with 6 colors.
+    bool runSixColoring() {
+
+      typename Graph::template NodeMap<int> heap_index(_graph, -1);
+      BucketHeap<typename Graph::template NodeMap<int> > heap(heap_index);
+
+      for (NodeIt n(_graph); n != INVALID; ++n) {
+        _color_map[n] = -2;
+        heap.push(n, countOutArcs(_graph, n));
+      }
+
+      std::vector<Node> order;
+
+      while (!heap.empty()) {
+        Node n = heap.top();
+        heap.pop();
+        _color_map[n] = -1;
+        order.push_back(n);
+        for (OutArcIt e(_graph, n); e != INVALID; ++e) {
+          Node t = _graph.runningNode(e);
+          if (_color_map[t] == -2) {
+            heap.decrease(t, heap[t] - 1);
+          }
+        }
+      }
+
+      for (int i = order.size() - 1; i >= 0; --i) {
+        std::vector<bool> forbidden(6, false);
+        for (OutArcIt e(_graph, order[i]); e != INVALID; ++e) {
+          Node t = _graph.runningNode(e);
+          if (_color_map[t] != -1) {
+            forbidden[_color_map[t]] = true;
+          }
+        }
+               for (int k = 0; k < 6; ++k) {
+          if (!forbidden[k]) {
+            _color_map[order[i]] = k;
+            break;
+          }
+        }
+        if (_color_map[order[i]] == -1) {
+          return false;
+        }
+      }
+      return true;
+    }
+
+  private:
+
+    bool recolor(const Node& u, const Node& v) {
+      int ucolor = _color_map[u];
+      int vcolor = _color_map[v];
+      typedef _planarity_bits::KempeFilter<IndexMap> KempeFilter;
+      KempeFilter filter(_color_map, ucolor, vcolor);
+
+      typedef FilterNodes<const Graph, const KempeFilter> KempeGraph;
+      KempeGraph kempe_graph(_graph, filter);
+
+      std::vector<Node> comp;
+      Bfs<KempeGraph> bfs(kempe_graph);
+      bfs.init();
+      bfs.addSource(u);
+      while (!bfs.emptyQueue()) {
+        Node n = bfs.nextNode();
+        if (n == v) return false;
+        comp.push_back(n);
+        bfs.processNextNode();
+      }
+
+      int scolor = ucolor + vcolor;
+      for (int i = 0; i < static_cast<int>(comp.size()); ++i) {
+        _color_map[comp[i]] = scolor - _color_map[comp[i]];
+      }
+
+      return true;
+    }
+
+    template <typename EmbeddingMap>
+    void kempeRecoloring(const Node& node, const EmbeddingMap& embedding) {
+      std::vector<Node> nodes;
+      nodes.reserve(4);
+
+      for (Arc e = OutArcIt(_graph, node); e != INVALID; e = embedding[e]) {
+        Node t = _graph.target(e);
+        if (_color_map[t] != -1) {
+          nodes.push_back(t);
+          if (nodes.size() == 4) break;
+        }
+      }
+
+      int color = _color_map[nodes[0]];
+      if (recolor(nodes[0], nodes[2])) {
+        _color_map[node] = color;
+      } else {
+        color = _color_map[nodes[1]];
+        recolor(nodes[1], nodes[3]);
+        _color_map[node] = color;
+      }
+    }
+
+  public:
+
+    /// \brief Calculates a coloring with at most five colors
+    ///
+    /// This function calculates a coloring with at most five
+    /// colors. The worst case time complexity of this variant is
+    /// quadratic in the size of the graph.
+    template <typename EmbeddingMap>
+    void runFiveColoring(const EmbeddingMap& embedding) {
+
+      typename Graph::template NodeMap<int> heap_index(_graph, -1);
+      BucketHeap<typename Graph::template NodeMap<int> > heap(heap_index);
+
+      for (NodeIt n(_graph); n != INVALID; ++n) {
+        _color_map[n] = -2;
+        heap.push(n, countOutArcs(_graph, n));
+      }
+
+      std::vector<Node> order;
+
+      while (!heap.empty()) {
+        Node n = heap.top();
+        heap.pop();
+        _color_map[n] = -1;
+        order.push_back(n);
+        for (OutArcIt e(_graph, n); e != INVALID; ++e) {
+          Node t = _graph.runningNode(e);
+          if (_color_map[t] == -2) {
+            heap.decrease(t, heap[t] - 1);
+          }
+        }
+      }
+
+      for (int i = order.size() - 1; i >= 0; --i) {
+        std::vector<bool> forbidden(5, false);
+        for (OutArcIt e(_graph, order[i]); e != INVALID; ++e) {
+          Node t = _graph.runningNode(e);
+          if (_color_map[t] != -1) {
+            forbidden[_color_map[t]] = true;
+          }
+        }
+        for (int k = 0; k < 5; ++k) {
+          if (!forbidden[k]) {
+            _color_map[order[i]] = k;
+            break;
+          }
+        }
+        if (_color_map[order[i]] == -1) {
+          kempeRecoloring(order[i], embedding);
+        }
+      }
+    }
+
+    /// \brief Calculates a coloring with at most five colors
+    ///
+    /// This function calculates a coloring with at most five
+    /// colors. The worst case time complexity of this variant is
+    /// quadratic in the size of the graph.
+    /// \return %True when the graph is planar.
+    bool runFiveColoring() {
+      PlanarEmbedding<Graph> pe(_graph);
+      if (!pe.run()) return false;
+
+      runFiveColoring(pe.embeddingMap());
+      return true;
+    }
+
+  private:
+
+    const Graph& _graph;
+    IndexMap _color_map;
+    Palette _palette;
+  };
+
+}
+
+#endif