/* -*- mode: C++; indent-tabs-mode: nil; -*-

  *

  * This file is a part of LEMON, a generic C++ optimization library.

  *

  * Copyright (C) 2003-2008

  * 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_TOPOLOGY_H

 #define LEMON_TOPOLOGY_H

 #include <lemon/dfs.h>

 #include <lemon/bfs.h>

 #include <lemon/core.h>

 #include <lemon/maps.h>

 #include <lemon/adaptors.h>

 #include <lemon/concepts/digraph.h>

 #include <lemon/concepts/graph.h>

 #include <lemon/concept_check.h>

 #include <stack>

 #include <functional>

 /// \ingroup connectivity

 /// \file

 /// \brief Connectivity algorithms

 ///

 /// Connectivity algorithms

 namespace lemon {

   /// \ingroup connectivity

   ///

   /// \brief Check whether the given undirected graph is connected.

   ///

   /// Check whether the given undirected graph is connected.

   /// \param graph The undirected graph.

   /// \return %True when there is path between any two nodes in the graph.

   /// \note By definition, the empty graph is connected.

   template <typename Graph>

   bool connected(const Graph& graph) {

     checkConcept<concepts::Graph, Graph>();

     typedef typename Graph::NodeIt NodeIt;

     if (NodeIt(graph) == INVALID) return true;

     Dfs<Graph> dfs(graph);

     dfs.run(NodeIt(graph));

     for (NodeIt it(graph); it != INVALID; ++it) {

       if (!dfs.reached(it)) {

         return false;

       }

     }

     return true;

   }

   /// \ingroup connectivity

   ///

   /// \brief Count the number of connected components of an undirected graph

   ///

   /// Count the number of connected components of an undirected graph

   ///

   /// \param graph The graph. It must be undirected.

   /// \return The number of components

   /// \note By definition, the empty graph consists

   /// of zero connected components.

   template <typename Graph>

   int countConnectedComponents(const Graph &graph) {

     checkConcept<concepts::Graph, Graph>();

     typedef typename Graph::Node Node;

     typedef typename Graph::Arc Arc;

     typedef NullMap<Node, Arc> PredMap;

     typedef NullMap<Node, int> DistMap;

     int compNum = 0;

     typename Bfs<Graph>::

       template SetPredMap<PredMap>::

       template SetDistMap<DistMap>::

       Create bfs(graph);

     PredMap predMap;

     bfs.predMap(predMap);

     DistMap distMap;

     bfs.distMap(distMap);

     bfs.init();

     for(typename Graph::NodeIt n(graph); n != INVALID; ++n) {

       if (!bfs.reached(n)) {

         bfs.addSource(n);

         bfs.start();

         ++compNum;

       }

     }

     return compNum;

   }

   /// \ingroup connectivity

   ///

   /// \brief Find the connected components of an undirected graph

   ///

   /// Find the connected components of an undirected graph.

   ///

   /// \param graph The graph. It must be undirected.

   /// \retval compMap A writable node map. The values will be set from 0 to

   /// the number of the connected components minus one. Each values of the map

   /// will be set exactly once, the values of a certain component will be

   /// set continuously.

   /// \return The number of components

   ///

   template <class Graph, class NodeMap>

   int connectedComponents(const Graph &graph, NodeMap &compMap) {

     checkConcept<concepts::Graph, Graph>();

     typedef typename Graph::Node Node;

     typedef typename Graph::Arc Arc;

     checkConcept<concepts::WriteMap<Node, int>, NodeMap>();

     typedef NullMap<Node, Arc> PredMap;

     typedef NullMap<Node, int> DistMap;

     int compNum = 0;

     typename Bfs<Graph>::

       template SetPredMap<PredMap>::

       template SetDistMap<DistMap>::

       Create bfs(graph);

     PredMap predMap;

     bfs.predMap(predMap);

     DistMap distMap;

     bfs.distMap(distMap);

     bfs.init();

     for(typename Graph::NodeIt n(graph); n != INVALID; ++n) {

       if(!bfs.reached(n)) {

         bfs.addSource(n);

         while (!bfs.emptyQueue()) {

           compMap.set(bfs.nextNode(), compNum);

           bfs.processNextNode();

         }

         ++compNum;

       }

     }

     return compNum;

   }

   namespace _topology_bits {

     template <typename Digraph, typename Iterator >

     struct LeaveOrderVisitor : public DfsVisitor<Digraph> {

     public:

       typedef typename Digraph::Node Node;

       LeaveOrderVisitor(Iterator it) : _it(it) {}

       void leave(const Node& node) {

         *(_it++) = node;

       }

     private:

       Iterator _it;

     };

     template <typename Digraph, typename Map>

     struct FillMapVisitor : public DfsVisitor<Digraph> {

     public:

       typedef typename Digraph::Node Node;

       typedef typename Map::Value Value;

       FillMapVisitor(Map& map, Value& value)

         : _map(map), _value(value) {}

       void reach(const Node& node) {

         _map.set(node, _value);

       }

     private:

       Map& _map;

       Value& _value;

     };

     template <typename Digraph, typename ArcMap>

     struct StronglyConnectedCutEdgesVisitor : public DfsVisitor<Digraph> {

     public:

       typedef typename Digraph::Node Node;

       typedef typename Digraph::Arc Arc;

       StronglyConnectedCutEdgesVisitor(const Digraph& digraph,

                                        ArcMap& cutMap,

                                        int& cutNum)

         : _digraph(digraph), _cutMap(cutMap), _cutNum(cutNum),

           _compMap(digraph), _num(0) {

       }

       void stop(const Node&) {

         ++_num;

       }

       void reach(const Node& node) {

         _compMap.set(node, _num);

       }

       void examine(const Arc& arc) {

          if (_compMap[_digraph.source(arc)] !=

              _compMap[_digraph.target(arc)]) {

            _cutMap.set(arc, true);

            ++_cutNum;

          }

       }

     private:

       const Digraph& _digraph;

       ArcMap& _cutMap;

       int& _cutNum;

       typename Digraph::template NodeMap<int> _compMap;

       int _num;

     };

   }

   /// \ingroup connectivity

   ///

   /// \brief Check whether the given directed graph is strongly connected.

   ///

   /// Check whether the given directed graph is strongly connected. The

   /// graph is strongly connected when any two nodes of the graph are

   /// connected with directed paths in both direction.

   /// \return %False when the graph is not strongly connected.

   /// \see connected

   ///

   /// \note By definition, the empty graph is strongly connected.

   template <typename Digraph>

   bool stronglyConnected(const Digraph& digraph) {

     checkConcept<concepts::Digraph, Digraph>();

     typedef typename Digraph::Node Node;

     typedef typename Digraph::NodeIt NodeIt;

     typename Digraph::Node source = NodeIt(digraph);

     if (source == INVALID) return true;

     using namespace _topology_bits;

     typedef DfsVisitor<Digraph> Visitor;

     Visitor visitor;

     DfsVisit<Digraph, Visitor> dfs(digraph, visitor);

     dfs.init();

     dfs.addSource(source);

     dfs.start();

     for (NodeIt it(digraph); it != INVALID; ++it) {

       if (!dfs.reached(it)) {

         return false;

       }

     }

     typedef ReverseDigraph<const Digraph> RDigraph;

     RDigraph rdigraph(digraph);

     typedef DfsVisitor<Digraph> RVisitor;

     RVisitor rvisitor;

     DfsVisit<RDigraph, RVisitor> rdfs(rdigraph, rvisitor);

     rdfs.init();

     rdfs.addSource(source);

     rdfs.start();

     for (NodeIt it(rdigraph); it != INVALID; ++it) {

       if (!rdfs.reached(it)) {

         return false;

       }

     }

     return true;

   }

   /// \ingroup connectivity

   ///

   /// \brief Count the strongly connected components of a directed graph

   ///

   /// Count the strongly connected components of a directed graph.

   /// The strongly connected components are the classes of an

   /// equivalence relation on the nodes of the graph. Two nodes are in

   /// the same class if they are connected with directed paths in both

   /// direction.

   ///

   /// \param graph The graph.

   /// \return The number of components

   /// \note By definition, the empty graph has zero

   /// strongly connected components.

   template <typename Digraph>

   int countStronglyConnectedComponents(const Digraph& digraph) {

     checkConcept<concepts::Digraph, Digraph>();

     using namespace _topology_bits;

     typedef typename Digraph::Node Node;

     typedef typename Digraph::Arc Arc;

     typedef typename Digraph::NodeIt NodeIt;

     typedef typename Digraph::ArcIt ArcIt;

     typedef std::vector<Node> Container;

     typedef typename Container::iterator Iterator;

     Container nodes(countNodes(digraph));

     typedef LeaveOrderVisitor<Digraph, Iterator> Visitor;

     Visitor visitor(nodes.begin());

     DfsVisit<Digraph, Visitor> dfs(digraph, visitor);

     dfs.init();

     for (NodeIt it(digraph); it != INVALID; ++it) {

       if (!dfs.reached(it)) {

         dfs.addSource(it);

         dfs.start();

       }

     }

     typedef typename Container::reverse_iterator RIterator;

     typedef ReverseDigraph<const Digraph> RDigraph;

     RDigraph rdigraph(digraph);

     typedef DfsVisitor<Digraph> RVisitor;

     RVisitor rvisitor;

     DfsVisit<RDigraph, RVisitor> rdfs(rdigraph, rvisitor);

     int compNum = 0;

     rdfs.init();

     for (RIterator it = nodes.rbegin(); it != nodes.rend(); ++it) {

       if (!rdfs.reached(*it)) {

         rdfs.addSource(*it);

         rdfs.start();

         ++compNum;

       }

     }

     return compNum;

   }

   /// \ingroup connectivity

   ///

   /// \brief Find the strongly connected components of a directed graph

   ///

   /// Find the strongly connected components of a directed graph.  The

   /// strongly connected components are the classes of an equivalence

   /// relation on the nodes of the graph. Two nodes are in

   /// relationship when there are directed paths between them in both

   /// direction. In addition, the numbering of components will satisfy

   /// that there is no arc going from a higher numbered component to

   /// a lower.

   ///

   /// \param digraph The digraph.

   /// \retval compMap A writable node map. The values will be set from 0 to

   /// the number of the strongly connected components minus one. Each value

   /// of the map will be set exactly once, the values of a certain component

   /// will be set continuously.

   /// \return The number of components

   ///

   template <typename Digraph, typename NodeMap>

   int stronglyConnectedComponents(const Digraph& digraph, NodeMap& compMap) {

     checkConcept<concepts::Digraph, Digraph>();

     typedef typename Digraph::Node Node;

     typedef typename Digraph::NodeIt NodeIt;

     checkConcept<concepts::WriteMap<Node, int>, NodeMap>();

     using namespace _topology_bits;

     typedef std::vector<Node> Container;

     typedef typename Container::iterator Iterator;

     Container nodes(countNodes(digraph));

     typedef LeaveOrderVisitor<Digraph, Iterator> Visitor;

     Visitor visitor(nodes.begin());

     DfsVisit<Digraph, Visitor> dfs(digraph, visitor);

     dfs.init();

     for (NodeIt it(digraph); it != INVALID; ++it) {

       if (!dfs.reached(it)) {

         dfs.addSource(it);

         dfs.start();

       }

     }

     typedef typename Container::reverse_iterator RIterator;

     typedef ReverseDigraph<const Digraph> RDigraph;

     RDigraph rdigraph(digraph);

     int compNum = 0;

     typedef FillMapVisitor<RDigraph, NodeMap> RVisitor;

     RVisitor rvisitor(compMap, compNum);

     DfsVisit<RDigraph, RVisitor> rdfs(rdigraph, rvisitor);

     rdfs.init();

     for (RIterator it = nodes.rbegin(); it != nodes.rend(); ++it) {

       if (!rdfs.reached(*it)) {

         rdfs.addSource(*it);

         rdfs.start();

         ++compNum;

       }

     }

     return compNum;

   }

   /// \ingroup connectivity

   ///

   /// \brief Find the cut arcs of the strongly connected components.

   ///

   /// Find the cut arcs of the strongly connected components.

   /// The strongly connected components are the classes of an equivalence

   /// relation on the nodes of the graph. Two nodes are in relationship

   /// when there are directed paths between them in both direction.

   /// The strongly connected components are separated by the cut arcs.

   ///

   /// \param graph The graph.

   /// \retval cutMap A writable node map. The values will be set true when the

   /// arc is a cut arc.

   ///

   /// \return The number of cut arcs

   template <typename Digraph, typename ArcMap>

   int stronglyConnectedCutArcs(const Digraph& graph, ArcMap& cutMap) {

     checkConcept<concepts::Digraph, Digraph>();

     typedef typename Digraph::Node Node;

     typedef typename Digraph::Arc Arc;

     typedef typename Digraph::NodeIt NodeIt;

     checkConcept<concepts::WriteMap<Arc, bool>, ArcMap>();

     using namespace _topology_bits;

     typedef std::vector<Node> Container;

     typedef typename Container::iterator Iterator;

     Container nodes(countNodes(graph));

     typedef LeaveOrderVisitor<Digraph, Iterator> Visitor;

     Visitor visitor(nodes.begin());

     DfsVisit<Digraph, Visitor> dfs(graph, visitor);

     dfs.init();

     for (NodeIt it(graph); it != INVALID; ++it) {

       if (!dfs.reached(it)) {

         dfs.addSource(it);

         dfs.start();

       }

     }

     typedef typename Container::reverse_iterator RIterator;

     typedef ReverseDigraph<const Digraph> RDigraph;

     RDigraph rgraph(graph);

     int cutNum = 0;

     typedef StronglyConnectedCutEdgesVisitor<RDigraph, ArcMap> RVisitor;

     RVisitor rvisitor(rgraph, cutMap, cutNum);

     DfsVisit<RDigraph, RVisitor> rdfs(rgraph, rvisitor);

     rdfs.init();

     for (RIterator it = nodes.rbegin(); it != nodes.rend(); ++it) {

       if (!rdfs.reached(*it)) {

         rdfs.addSource(*it);

         rdfs.start();

       }

     }

     return cutNum;

   }

   namespace _topology_bits {

     template <typename Digraph>

     class CountBiNodeConnectedComponentsVisitor : public DfsVisitor<Digraph> {

     public:

       typedef typename Digraph::Node Node;

       typedef typename Digraph::Arc Arc;

       typedef typename Digraph::Edge Edge;

       CountBiNodeConnectedComponentsVisitor(const Digraph& graph, int &compNum)

         : _graph(graph), _compNum(compNum),

           _numMap(graph), _retMap(graph), _predMap(graph), _num(0) {}

       void start(const Node& node) {

         _predMap.set(node, INVALID);

       }

       void reach(const Node& node) {

         _numMap.set(node, _num);

         _retMap.set(node, _num);

         ++_num;

       }

       void discover(const Arc& edge) {

         _predMap.set(_graph.target(edge), _graph.source(edge));

       }

       void examine(const Arc& edge) {

         if (_graph.source(edge) == _graph.target(edge) &&

             _graph.direction(edge)) {

           ++_compNum;

           return;

         }

         if (_predMap[_graph.source(edge)] == _graph.target(edge)) {

           return;

         }

         if (_retMap[_graph.source(edge)] > _numMap[_graph.target(edge)]) {

           _retMap.set(_graph.source(edge), _numMap[_graph.target(edge)]);

         }

       }

       void backtrack(const Arc& edge) {

         if (_retMap[_graph.source(edge)] > _retMap[_graph.target(edge)]) {

           _retMap.set(_graph.source(edge), _retMap[_graph.target(edge)]);

         }

         if (_numMap[_graph.source(edge)] <= _retMap[_graph.target(edge)]) {

           ++_compNum;

         }

       }

     private:

       const Digraph& _graph;

       int& _compNum;

       typename Digraph::template NodeMap<int> _numMap;

       typename Digraph::template NodeMap<int> _retMap;

       typename Digraph::template NodeMap<Node> _predMap;

       int _num;

     };

     template <typename Digraph, typename ArcMap>

     class BiNodeConnectedComponentsVisitor : public DfsVisitor<Digraph> {

     public:

       typedef typename Digraph::Node Node;

       typedef typename Digraph::Arc Arc;

       typedef typename Digraph::Edge Edge;

       BiNodeConnectedComponentsVisitor(const Digraph& graph,

                                        ArcMap& compMap, int &compNum)

         : _graph(graph), _compMap(compMap), _compNum(compNum),

           _numMap(graph), _retMap(graph), _predMap(graph), _num(0) {}

       void start(const Node& node) {

         _predMap.set(node, INVALID);

       }

       void reach(const Node& node) {

         _numMap.set(node, _num);

         _retMap.set(node, _num);

         ++_num;

       }

       void discover(const Arc& edge) {

         Node target = _graph.target(edge);

         _predMap.set(target, edge);

         _edgeStack.push(edge);

       }

       void examine(const Arc& edge) {

         Node source = _graph.source(edge);

         Node target = _graph.target(edge);

         if (source == target && _graph.direction(edge)) {

           _compMap.set(edge, _compNum);

           ++_compNum;

           return;

         }

         if (_numMap[target] < _numMap[source]) {

           if (_predMap[source] != _graph.oppositeArc(edge)) {

             _edgeStack.push(edge);

           }

         }

         if (_predMap[source] != INVALID &&

             target == _graph.source(_predMap[source])) {

           return;

         }

         if (_retMap[source] > _numMap[target]) {

           _retMap.set(source, _numMap[target]);

         }

       }

       void backtrack(const Arc& edge) {

         Node source = _graph.source(edge);

         Node target = _graph.target(edge);

         if (_retMap[source] > _retMap[target]) {

           _retMap.set(source, _retMap[target]);

         }

         if (_numMap[source] <= _retMap[target]) {

           while (_edgeStack.top() != edge) {

             _compMap.set(_edgeStack.top(), _compNum);

             _edgeStack.pop();

           }

           _compMap.set(edge, _compNum);

           _edgeStack.pop();

           ++_compNum;

         }

       }

     private:

       const Digraph& _graph;

       ArcMap& _compMap;

       int& _compNum;

       typename Digraph::template NodeMap<int> _numMap;

       typename Digraph::template NodeMap<int> _retMap;

       typename Digraph::template NodeMap<Arc> _predMap;

       std::stack<Edge> _edgeStack;

       int _num;

     };

     template <typename Digraph, typename NodeMap>

     class BiNodeConnectedCutNodesVisitor : public DfsVisitor<Digraph> {

     public:

       typedef typename Digraph::Node Node;

       typedef typename Digraph::Arc Arc;

       typedef typename Digraph::Edge Edge;

       BiNodeConnectedCutNodesVisitor(const Digraph& graph, NodeMap& cutMap,

                                      int& cutNum)

         : _graph(graph), _cutMap(cutMap), _cutNum(cutNum),

           _numMap(graph), _retMap(graph), _predMap(graph), _num(0) {}

       void start(const Node& node) {

         _predMap.set(node, INVALID);

         rootCut = false;

       }

       void reach(const Node& node) {

         _numMap.set(node, _num);

         _retMap.set(node, _num);

         ++_num;

       }

       void discover(const Arc& edge) {

         _predMap.set(_graph.target(edge), _graph.source(edge));

       }

       void examine(const Arc& edge) {

         if (_graph.source(edge) == _graph.target(edge) &&

             _graph.direction(edge)) {

           if (!_cutMap[_graph.source(edge)]) {

             _cutMap.set(_graph.source(edge), true);

             ++_cutNum;

           }

           return;

         }

         if (_predMap[_graph.source(edge)] == _graph.target(edge)) return;

         if (_retMap[_graph.source(edge)] > _numMap[_graph.target(edge)]) {

           _retMap.set(_graph.source(edge), _numMap[_graph.target(edge)]);

         }

       }

       void backtrack(const Arc& edge) {

         if (_retMap[_graph.source(edge)] > _retMap[_graph.target(edge)]) {

           _retMap.set(_graph.source(edge), _retMap[_graph.target(edge)]);

         }

         if (_numMap[_graph.source(edge)] <= _retMap[_graph.target(edge)]) {

           if (_predMap[_graph.source(edge)] != INVALID) {

             if (!_cutMap[_graph.source(edge)]) {

               _cutMap.set(_graph.source(edge), true);

               ++_cutNum;

             }

           } else if (rootCut) {

             if (!_cutMap[_graph.source(edge)]) {

               _cutMap.set(_graph.source(edge), true);

               ++_cutNum;

             }

           } else {

             rootCut = true;

           }

         }

       }

     private:

       const Digraph& _graph;

       NodeMap& _cutMap;

       int& _cutNum;

       typename Digraph::template NodeMap<int> _numMap;

       typename Digraph::template NodeMap<int> _retMap;

       typename Digraph::template NodeMap<Node> _predMap;

       std::stack<Edge> _edgeStack;

       int _num;

       bool rootCut;

     };

   }

   template <typename Graph>

   int countBiNodeConnectedComponents(const Graph& graph);

   /// \ingroup connectivity

   ///

   /// \brief Checks the graph is bi-node-connected.

   ///

   /// This function checks that the undirected graph is bi-node-connected

   /// graph. The graph is bi-node-connected if any two undirected edge is

   /// on same circle.

   ///

   /// \param graph The graph.

   /// \return %True when the graph bi-node-connected.

   template <typename Graph>

   bool biNodeConnected(const Graph& graph) {

     return countBiNodeConnectedComponents(graph) <= 1;

   }

   /// \ingroup connectivity

   ///

   /// \brief Count the biconnected components.

   ///

   /// This function finds the bi-node-connected components in an undirected

   /// graph. The biconnected components are the classes of an equivalence

   /// relation on the undirected edges. Two undirected edge is in relationship

   /// when they are on same circle.

   ///

   /// \param graph The graph.

   /// \return The number of components.

   template <typename Graph>

   int countBiNodeConnectedComponents(const Graph& graph) {

     checkConcept<concepts::Graph, Graph>();

     typedef typename Graph::NodeIt NodeIt;

     using namespace _topology_bits;

     typedef CountBiNodeConnectedComponentsVisitor<Graph> Visitor;

     int compNum = 0;

     Visitor visitor(graph, compNum);

     DfsVisit<Graph, Visitor> dfs(graph, visitor);

     dfs.init();

     for (NodeIt it(graph); it != INVALID; ++it) {

       if (!dfs.reached(it)) {

         dfs.addSource(it);

         dfs.start();

       }

     }

     return compNum;

   }

   /// \ingroup connectivity

   ///

   /// \brief Find the bi-node-connected components.

   ///

   /// This function finds the bi-node-connected components in an undirected

   /// graph. The bi-node-connected components are the classes of an equivalence

   /// relation on the undirected edges. Two undirected edge are in relationship

   /// when they are on same circle.

   ///

   /// \param graph The graph.

   /// \retval compMap A writable uedge map. The values will be set from 0

   /// to the number of the biconnected components minus one. Each values

   /// of the map will be set exactly once, the values of a certain component

   /// will be set continuously.

   /// \return The number of components.

   ///

   template <typename Graph, typename EdgeMap>

   int biNodeConnectedComponents(const Graph& graph,

                                 EdgeMap& compMap) {

     checkConcept<concepts::Graph, Graph>();

     typedef typename Graph::NodeIt NodeIt;

     typedef typename Graph::Edge Edge;

     checkConcept<concepts::WriteMap<Edge, int>, EdgeMap>();

     using namespace _topology_bits;

     typedef BiNodeConnectedComponentsVisitor<Graph, EdgeMap> Visitor;

     int compNum = 0;

     Visitor visitor(graph, compMap, compNum);

     DfsVisit<Graph, Visitor> dfs(graph, visitor);

     dfs.init();

     for (NodeIt it(graph); it != INVALID; ++it) {

       if (!dfs.reached(it)) {

         dfs.addSource(it);

         dfs.start();

       }

     }

     return compNum;

   }

   /// \ingroup connectivity

   ///

   /// \brief Find the bi-node-connected cut nodes.

   ///

   /// This function finds the bi-node-connected cut nodes in an undirected

   /// graph. The bi-node-connected components are the classes of an equivalence

   /// relation on the undirected edges. Two undirected edges are in

   /// relationship when they are on same circle. The biconnected components

   /// are separted by nodes which are the cut nodes of the components.

   ///

   /// \param graph The graph.

   /// \retval cutMap A writable edge map. The values will be set true when

   /// the node separate two or more components.

   /// \return The number of the cut nodes.

   template <typename Graph, typename NodeMap>

   int biNodeConnectedCutNodes(const Graph& graph, NodeMap& cutMap) {

     checkConcept<concepts::Graph, Graph>();

     typedef typename Graph::Node Node;

     typedef typename Graph::NodeIt NodeIt;

     checkConcept<concepts::WriteMap<Node, bool>, NodeMap>();

     using namespace _topology_bits;

     typedef BiNodeConnectedCutNodesVisitor<Graph, NodeMap> Visitor;

     int cutNum = 0;

     Visitor visitor(graph, cutMap, cutNum);

     DfsVisit<Graph, Visitor> dfs(graph, visitor);

     dfs.init();

     for (NodeIt it(graph); it != INVALID; ++it) {

       if (!dfs.reached(it)) {

         dfs.addSource(it);

         dfs.start();

       }

     }

     return cutNum;

   }

   namespace _topology_bits {

     template <typename Digraph>

     class CountBiEdgeConnectedComponentsVisitor : public DfsVisitor<Digraph> {

     public:

       typedef typename Digraph::Node Node;

       typedef typename Digraph::Arc Arc;

       typedef typename Digraph::Edge Edge;

       CountBiEdgeConnectedComponentsVisitor(const Digraph& graph, int &compNum)

         : _graph(graph), _compNum(compNum),

           _numMap(graph), _retMap(graph), _predMap(graph), _num(0) {}

       void start(const Node& node) {

         _predMap.set(node, INVALID);

       }

       void reach(const Node& node) {

         _numMap.set(node, _num);

         _retMap.set(node, _num);

         ++_num;

       }

       void leave(const Node& node) {

         if (_numMap[node] <= _retMap[node]) {

           ++_compNum;

         }

       }

       void discover(const Arc& edge) {

         _predMap.set(_graph.target(edge), edge);

       }

       void examine(const Arc& edge) {

         if (_predMap[_graph.source(edge)] == _graph.oppositeArc(edge)) {

           return;

         }

         if (_retMap[_graph.source(edge)] > _retMap[_graph.target(edge)]) {

           _retMap.set(_graph.source(edge), _retMap[_graph.target(edge)]);

         }

       }

       void backtrack(const Arc& edge) {

         if (_retMap[_graph.source(edge)] > _retMap[_graph.target(edge)]) {

           _retMap.set(_graph.source(edge), _retMap[_graph.target(edge)]);

         }

       }

     private:

       const Digraph& _graph;

       int& _compNum;

       typename Digraph::template NodeMap<int> _numMap;

       typename Digraph::template NodeMap<int> _retMap;

       typename Digraph::template NodeMap<Arc> _predMap;

       int _num;

     };

     template <typename Digraph, typename NodeMap>

     class BiEdgeConnectedComponentsVisitor : public DfsVisitor<Digraph> {

     public:

       typedef typename Digraph::Node Node;

       typedef typename Digraph::Arc Arc;

       typedef typename Digraph::Edge Edge;

       BiEdgeConnectedComponentsVisitor(const Digraph& graph,

                                        NodeMap& compMap, int &compNum)

         : _graph(graph), _compMap(compMap), _compNum(compNum),

           _numMap(graph), _retMap(graph), _predMap(graph), _num(0) {}

       void start(const Node& node) {

         _predMap.set(node, INVALID);

       }

       void reach(const Node& node) {

         _numMap.set(node, _num);

         _retMap.set(node, _num);

         _nodeStack.push(node);

         ++_num;

       }

       void leave(const Node& node) {

         if (_numMap[node] <= _retMap[node]) {

           while (_nodeStack.top() != node) {

             _compMap.set(_nodeStack.top(), _compNum);

             _nodeStack.pop();

           }

           _compMap.set(node, _compNum);

           _nodeStack.pop();

           ++_compNum;

         }

       }

       void discover(const Arc& edge) {

         _predMap.set(_graph.target(edge), edge);

       }

       void examine(const Arc& edge) {

         if (_predMap[_graph.source(edge)] == _graph.oppositeArc(edge)) {

           return;

         }

         if (_retMap[_graph.source(edge)] > _retMap[_graph.target(edge)]) {

           _retMap.set(_graph.source(edge), _retMap[_graph.target(edge)]);

         }

       }

       void backtrack(const Arc& edge) {

         if (_retMap[_graph.source(edge)] > _retMap[_graph.target(edge)]) {

           _retMap.set(_graph.source(edge), _retMap[_graph.target(edge)]);

         }

       }

     private:

       const Digraph& _graph;

       NodeMap& _compMap;

       int& _compNum;

       typename Digraph::template NodeMap<int> _numMap;

       typename Digraph::template NodeMap<int> _retMap;

       typename Digraph::template NodeMap<Arc> _predMap;

       std::stack<Node> _nodeStack;

       int _num;

     };

     template <typename Digraph, typename ArcMap>

     class BiEdgeConnectedCutEdgesVisitor : public DfsVisitor<Digraph> {

     public:

       typedef typename Digraph::Node Node;

       typedef typename Digraph::Arc Arc;

       typedef typename Digraph::Edge Edge;

       BiEdgeConnectedCutEdgesVisitor(const Digraph& graph,

                                      ArcMap& cutMap, int &cutNum)

         : _graph(graph), _cutMap(cutMap), _cutNum(cutNum),

           _numMap(graph), _retMap(graph), _predMap(graph), _num(0) {}

       void start(const Node& node) {

         _predMap[node] = INVALID;

       }

       void reach(const Node& node) {

         _numMap.set(node, _num);

         _retMap.set(node, _num);

         ++_num;

       }

       void leave(const Node& node) {

         if (_numMap[node] <= _retMap[node]) {

           if (_predMap[node] != INVALID) {

             _cutMap.set(_predMap[node], true);

             ++_cutNum;

           }

         }

       }

       void discover(const Arc& edge) {

         _predMap.set(_graph.target(edge), edge);

       }

       void examine(const Arc& edge) {

         if (_predMap[_graph.source(edge)] == _graph.oppositeArc(edge)) {

           return;

         }

         if (_retMap[_graph.source(edge)] > _retMap[_graph.target(edge)]) {

           _retMap.set(_graph.source(edge), _retMap[_graph.target(edge)]);

         }

       }

       void backtrack(const Arc& edge) {

         if (_retMap[_graph.source(edge)] > _retMap[_graph.target(edge)]) {

           _retMap.set(_graph.source(edge), _retMap[_graph.target(edge)]);

         }

       }

     private:

       const Digraph& _graph;

       ArcMap& _cutMap;

       int& _cutNum;

       typename Digraph::template NodeMap<int> _numMap;

       typename Digraph::template NodeMap<int> _retMap;

       typename Digraph::template NodeMap<Arc> _predMap;

       int _num;

     };

   }

   template <typename Graph>

   int countBiEdgeConnectedComponents(const Graph& graph);

   /// \ingroup connectivity

   ///

   /// \brief Checks that the graph is bi-edge-connected.

   ///

   /// This function checks that the graph is bi-edge-connected. The undirected

   /// graph is bi-edge-connected when any two nodes are connected with two

   /// edge-disjoint paths.

   ///

   /// \param graph The undirected graph.

   /// \return The number of components.

   template <typename Graph>

   bool biEdgeConnected(const Graph& graph) {

     return countBiEdgeConnectedComponents(graph) <= 1;

   }

   /// \ingroup connectivity

   ///

   /// \brief Count the bi-edge-connected components.

   ///

   /// This function count the bi-edge-connected components in an undirected

   /// graph. The bi-edge-connected components are the classes of an equivalence

   /// relation on the nodes. Two nodes are in relationship when they are

   /// connected with at least two edge-disjoint paths.

   ///

   /// \param graph The undirected graph.

   /// \return The number of components.

   template <typename Graph>

   int countBiEdgeConnectedComponents(const Graph& graph) {

     checkConcept<concepts::Graph, Graph>();

     typedef typename Graph::NodeIt NodeIt;

     using namespace _topology_bits;

     typedef CountBiEdgeConnectedComponentsVisitor<Graph> Visitor;

     int compNum = 0;

     Visitor visitor(graph, compNum);

     DfsVisit<Graph, Visitor> dfs(graph, visitor);

     dfs.init();

     for (NodeIt it(graph); it != INVALID; ++it) {

       if (!dfs.reached(it)) {

         dfs.addSource(it);

         dfs.start();

       }

     }

     return compNum;

   }

   /// \ingroup connectivity

   ///

   /// \brief Find the bi-edge-connected components.

   ///

   /// This function finds the bi-edge-connected components in an undirected

   /// graph. The bi-edge-connected components are the classes of an equivalence

   /// relation on the nodes. Two nodes are in relationship when they are

   /// connected at least two edge-disjoint paths.

   ///

   /// \param graph The graph.

   /// \retval compMap A writable node map. The values will be set from 0 to

   /// the number of the biconnected components minus one. Each values

   /// of the map will be set exactly once, the values of a certain component

   /// will be set continuously.

   /// \return The number of components.

   ///

   template <typename Graph, typename NodeMap>

   int biEdgeConnectedComponents(const Graph& graph, NodeMap& compMap) {

     checkConcept<concepts::Graph, Graph>();

     typedef typename Graph::NodeIt NodeIt;

     typedef typename Graph::Node Node;

     checkConcept<concepts::WriteMap<Node, int>, NodeMap>();

     using namespace _topology_bits;

     typedef BiEdgeConnectedComponentsVisitor<Graph, NodeMap> Visitor;

     int compNum = 0;

     Visitor visitor(graph, compMap, compNum);

     DfsVisit<Graph, Visitor> dfs(graph, visitor);

     dfs.init();

     for (NodeIt it(graph); it != INVALID; ++it) {

       if (!dfs.reached(it)) {

         dfs.addSource(it);

         dfs.start();

       }

     }

     return compNum;

   }

   /// \ingroup connectivity

   ///

   /// \brief Find the bi-edge-connected cut edges.

   ///

   /// This function finds the bi-edge-connected components in an undirected

   /// graph. The bi-edge-connected components are the classes of an equivalence

   /// relation on the nodes. Two nodes are in relationship when they are

   /// connected with at least two edge-disjoint paths. The bi-edge-connected

   /// components are separted by edges which are the cut edges of the

   /// components.

   ///

   /// \param graph The graph.

   /// \retval cutMap A writable node map. The values will be set true when the

   /// edge is a cut edge.

   /// \return The number of cut edges.

   template <typename Graph, typename EdgeMap>

   int biEdgeConnectedCutEdges(const Graph& graph, EdgeMap& cutMap) {

     checkConcept<concepts::Graph, Graph>();

     typedef typename Graph::NodeIt NodeIt;

     typedef typename Graph::Edge Edge;