/* -*- 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_CONNECTIVITY_H

 #define LEMON_CONNECTIVITY_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 graph_properties

 /// \file

 /// \brief Connectivity algorithms

 ///

 /// Connectivity algorithms

 namespace lemon {

   /// \ingroup graph_properties

   ///

   /// \brief Check whether an undirected graph is connected.

   ///

   /// This function checks whether the given undirected graph is connected,

   /// i.e. there is a path between any two nodes in the graph.

   ///

   /// \return \c true if the graph is connected.

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

   ///

   /// \see countConnectedComponents(), connectedComponents()

   /// \see stronglyConnected()

   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 graph_properties

   ///

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

   ///

   /// This function counts the number of connected components of the given

   /// undirected graph.

   ///

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

   /// on the nodes of an undirected graph. Two nodes are in the same class

   /// if they are connected with a path.

   ///

   /// \return The number of connected components.

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

   /// of zero connected components.

   ///

   /// \see connected(), connectedComponents()

   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 graph_properties

   ///

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

   ///

   /// This function finds the connected components of the given undirected

   /// graph.

   ///

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

   /// on the nodes of an undirected graph. Two nodes are in the same class

   /// if they are connected with a path.

   ///

   /// \image html connected_components.png

   /// \image latex connected_components.eps "Connected components" width=\textwidth

   ///

   /// \param graph The undirected graph.

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

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

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

   /// set continuously.

   /// \return The number of connected components.

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

   /// of zero connected components.

   ///

   /// \see connected(), countConnectedComponents()

   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 _connectivity_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 StronglyConnectedCutArcsVisitor : public DfsVisitor<Digraph> {

     public:

       typedef typename Digraph::Node Node;

       typedef typename Digraph::Arc Arc;

       StronglyConnectedCutArcsVisitor(const Digraph& digraph,

                                       ArcMap& cutMap,

                                       int& cutNum)

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

           _compMap(digraph, -1), _num(-1) {

       }

       void start(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 graph_properties

   ///

   /// \brief Check whether a directed graph is strongly connected.

   ///

   /// This function checks whether the given directed graph is strongly

   /// connected, i.e. any two nodes of the digraph are

   /// connected with directed paths in both direction.

   ///

   /// \return \c true if the digraph is strongly connected.

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

   /// 

   /// \see countStronglyConnectedComponents(), stronglyConnectedComponents()

   /// \see 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 _connectivity_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;

     typedef typename RDigraph::NodeIt RNodeIt;

     RDigraph rdigraph(digraph);

     typedef DfsVisitor<RDigraph> RVisitor;

     RVisitor rvisitor;

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

     rdfs.init();

     rdfs.addSource(source);

     rdfs.start();

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

       if (!rdfs.reached(it)) {

         return false;

       }

     }

     return true;

   }

   /// \ingroup graph_properties

   ///

   /// \brief Count the number of strongly connected components of a 

   /// directed graph

   ///

   /// This function counts the number of strongly connected components of

   /// the given directed graph.

   ///

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

   /// equivalence relation on the nodes of a digraph. Two nodes are in

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

   /// direction.

   ///

   /// \return The number of strongly connected components.

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

   /// strongly connected components.

   ///

   /// \see stronglyConnected(), stronglyConnectedComponents()

   template <typename Digraph>

   int countStronglyConnectedComponents(const Digraph& digraph) {

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

     using namespace _connectivity_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 graph_properties

   ///

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

   ///

   /// This function finds the strongly connected components of the given

   /// directed graph. In addition, the numbering of the components will

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

   /// to a lower one (i.e. it provides a topological order of the components).

   ///

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

   /// equivalence relation on the nodes of a digraph. Two nodes are in

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

   /// direction.

   ///

   /// \image html strongly_connected_components.png

   /// \image latex strongly_connected_components.eps "Strongly connected components" width=\textwidth

   ///

   /// \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, and the values of a certain

   /// component will be set continuously.

   /// \return The number of strongly connected components.

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

   /// strongly connected components.

   ///

   /// \see stronglyConnected(), countStronglyConnectedComponents()

   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 _connectivity_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 graph_properties

   ///

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

   ///

   /// This function finds the cut arcs of the strongly connected components

   /// of the given digraph.

   ///

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

   /// equivalence relation on the nodes of a digraph. Two nodes are in

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

   /// direction.

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

   ///

   /// \param digraph The digraph.

   /// \retval cutMap A writable arc map. The values will be set to \c true

   /// for the cut arcs (exactly once for each cut arc), and will not be

   /// changed for other arcs.

   /// \return The number of cut arcs.

   ///

   /// \see stronglyConnected(), stronglyConnectedComponents()

   template <typename Digraph, typename ArcMap>

   int stronglyConnectedCutArcs(const Digraph& digraph, 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 _connectivity_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 cutNum = 0;

     typedef StronglyConnectedCutArcsVisitor<RDigraph, ArcMap> RVisitor;

     RVisitor rvisitor(rdigraph, cutMap, cutNum);

     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();

       }

     }

     return cutNum;

   }

   namespace _connectivity_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 graph_properties

   ///

   /// \brief Check whether an undirected graph is bi-node-connected.

   ///

   /// This function checks whether the given undirected graph is

   /// bi-node-connected, i.e. a connected graph without articulation

   /// node.

   ///

   /// \return \c true if the graph bi-node-connected.

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

   ///

   /// \see countBiNodeConnectedComponents(), biNodeConnectedComponents()

   template <typename Graph>

   bool biNodeConnected(const Graph& graph) {

     bool hasNonIsolated = false, hasIsolated = false;

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

       if (typename Graph::OutArcIt(graph, n) == INVALID) {

         if (hasIsolated || hasNonIsolated) {

           return false;

         } else {

           hasIsolated = true;

         }

       } else {

         if (hasIsolated) {

           return false;

         } else {

           hasNonIsolated = true;

         }

       }

     }

     return countBiNodeConnectedComponents(graph) <= 1;

   }

   /// \ingroup graph_properties

   ///

   /// \brief Count the number of bi-node-connected components of an 

   /// undirected graph.

   ///

   /// This function counts the number of bi-node-connected components of

   /// the given undirected graph.

   ///

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

   /// relation on the edges of a undirected graph. Two edges are in the

   /// same class if they are on same circle.

   ///

   /// \return The number of bi-node-connected components.

   ///

   /// \see biNodeConnected(), biNodeConnectedComponents()

   template <typename Graph>

   int countBiNodeConnectedComponents(const Graph& graph) {

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

     typedef typename Graph::NodeIt NodeIt;

     using namespace _connectivity_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 graph_properties

   ///

   /// \brief Find the bi-node-connected components of an undirected graph.

   ///

   /// This function finds the bi-node-connected components of the given

   /// undirected graph.

   ///

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

   /// relation on the edges of a undirected graph. Two edges are in the

   /// same class if they are on same circle.

   ///

   /// \image html node_biconnected_components.png

   /// \image latex node_biconnected_components.eps "bi-node-connected components" width=\textwidth

   ///

   /// \param graph The undirected graph.

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

   /// to the number of the bi-node-connected components minus one. Each

   /// value of the map will be set exactly once, and the values of a 

   /// certain component will be set continuously.

   /// \return The number of bi-node-connected components.

   ///

   /// \see biNodeConnected(), countBiNodeConnectedComponents()

   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 _connectivity_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 graph_properties

   ///

   /// \brief Find the bi-node-connected cut nodes in an undirected graph.

   ///

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

   /// undirected graph.

   ///

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

   /// relation on the edges of a undirected graph. Two edges are in the

   /// same class if they are on same circle.

   /// The bi-node-connected components are separted by the cut nodes of

   /// the components.

   ///

   /// \param graph The undirected graph.

   /// \retval cutMap A writable node map. The values will be set to 

   /// \c true for the nodes that separate two or more components

   /// (exactly once for each cut node), and will not be changed for

   /// other nodes.

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

   ///

   /// \see biNodeConnected(), biNodeConnectedComponents()

   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 _connectivity_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 _connectivity_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 graph_properties

   ///

   /// \brief Check whether an undirected graph is bi-edge-connected.

   ///

   /// This function checks whether the given undirected graph is 

   /// bi-edge-connected, i.e. any two nodes are connected with at least

   /// two edge-disjoint paths.

   ///

   /// \return \c true if the graph is bi-edge-connected.

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

   ///

   /// \see countBiEdgeConnectedComponents(), biEdgeConnectedComponents()

   template <typename Graph>

   bool biEdgeConnected(const Graph& graph) {

     return countBiEdgeConnectedComponents(graph) <= 1;

   }

   /// \ingroup graph_properties

   ///

   /// \brief Count the number of bi-edge-connected components of an

   /// undirected graph.

   ///

   /// This function counts the number of bi-edge-connected components of

   /// the given undirected graph.

   ///

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

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

   /// same class if they are connected with at least two edge-disjoint

   /// paths.

   ///

   /// \return The number of bi-edge-connected components.

   ///

   /// \see biEdgeConnected(), biEdgeConnectedComponents()

   template <typename Graph>

   int countBiEdgeConnectedComponents(const Graph& graph) {

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

     typedef typename Graph::NodeIt NodeIt;

     using namespace _connectivity_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 graph_properties

   ///

   /// \brief Find the bi-edge-connected components of an undirected graph.

   ///

   /// This function finds the bi-edge-connected components of the given

   /// undirected graph.

   ///

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

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

   /// same class if they are connected with at least two edge-disjoint

   /// paths.

   ///

   /// \image html edge_biconnected_components.png

   /// \image latex edge_biconnected_components.eps "bi-edge-connected components" width=\textwidth

   ///

   /// \param graph The undirected graph.

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

   /// the number of the bi-edge-connected components minus one. Each value

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

   /// component will be set continuously.

   /// \return The number of bi-edge-connected components.

   ///

   /// \see biEdgeConnected(), countBiEdgeConnectedComponents()

   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 _connectivity_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 graph_properties

   ///

   /// \brief Find the bi-edge-connected cut edges in an undirected graph.

   ///

   /// This function finds the bi-edge-connected cut edges in the given

   /// undirected graph. 

   ///

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

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

   /// same class if they are connected with at least two edge-disjoint

   /// paths.

   /// The bi-edge-connected components are separted by the cut edges of

   /// the components.

   ///

   /// \param graph The undirected graph.

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

   /// for the cut edges (exactly once for each cut edge), and will not be

   /// changed for other edges.

   /// \return The number of cut edges.

   ///

   /// \see biEdgeConnected(), biEdgeConnectedComponents()

   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;

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

     using namespace _connectivity_bits;

     typedef BiEdgeConnectedCutEdgesVisitor<Graph, EdgeMap> 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 _connectivity_bits {

     template <typename Digraph, typename IntNodeMap>

     class TopologicalSortVisitor : public DfsVisitor<Digraph> {

     public:

       typedef typename Digraph::Node Node;

       typedef typename Digraph::Arc edge;

       TopologicalSortVisitor(IntNodeMap& order, int num)

         : _order(order), _num(num) {}

       void leave(const Node& node) {

         _order.set(node, --_num);

       }

     private:

       IntNodeMap& _order;

       int _num;

     };

   }

   /// \ingroup graph_properties

   ///

   /// \brief Check whether a digraph is DAG.

   ///

   /// This function checks whether the given digraph is DAG, i.e.

   /// \e Directed \e Acyclic \e Graph.

   /// \return \c true if there is no directed cycle in the digraph.

   /// \see acyclic()

   template <typename Digraph>

   bool dag(const Digraph& digraph) {

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

     typedef typename Digraph::Node Node;

     typedef typename Digraph::NodeIt NodeIt;

     typedef typename Digraph::Arc Arc;

     typedef typename Digraph::template NodeMap<bool> ProcessedMap;

     typename Dfs<Digraph>::template SetProcessedMap<ProcessedMap>::

       Create dfs(digraph);

     ProcessedMap processed(digraph);

     dfs.processedMap(processed);

     dfs.init();

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

       if (!dfs.reached(it)) {

         dfs.addSource(it);

         while (!dfs.emptyQueue()) {

           Arc arc = dfs.nextArc();

           Node target = digraph.target(arc);

           if (dfs.reached(target) && !processed[target]) {

             return false;

           }

           dfs.processNextArc();

         }

       }

     }

     return true;

   }

   /// \ingroup graph_properties

   ///

   /// \brief Sort the nodes of a DAG into topolgical order.

   ///

   /// This function sorts the nodes of the given acyclic digraph (DAG)

   /// into topolgical order.

   ///

   /// \param digraph The digraph, which must be DAG.

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

   /// the number of the nodes in the digraph minus one. Each value of the

   /// map will be set exactly once, and the values will be set descending

   /// order.

   ///

   /// \see dag(), checkedTopologicalSort()

   template <typename Digraph, typename NodeMap>

   void topologicalSort(const Digraph& digraph, NodeMap& order) {

     using namespace _connectivity_bits;

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

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

     typedef typename Digraph::Node Node;

     typedef typename Digraph::NodeIt NodeIt;

     typedef typename Digraph::Arc Arc;

     TopologicalSortVisitor<Digraph, NodeMap>

       visitor(order, countNodes(digraph));

     DfsVisit<Digraph, TopologicalSortVisitor<Digraph, NodeMap> >

       dfs(digraph, visitor);

     dfs.init();

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

       if (!dfs.reached(it)) {

         dfs.addSource(it);

         dfs.start();

       }

     }

   }

   /// \ingroup graph_properties

   ///

   /// \brief Sort the nodes of a DAG into topolgical order.

   ///

   /// This function sorts the nodes of the given acyclic digraph (DAG)

   /// into topolgical order and also checks whether the given digraph

   /// is DAG.

   ///

   /// \param digraph The digraph.

   /// \retval order A readable and writable node map. The values will be

   /// set from 0 to the number of the nodes in the digraph minus one. 

   /// Each value of the map will be set exactly once, and the values will

   /// be set descending order.

   /// \return \c false if the digraph is not DAG.

   ///

   /// \see dag(), topologicalSort()

   template <typename Digraph, typename NodeMap>

   bool checkedTopologicalSort(const Digraph& digraph, NodeMap& order) {

     using namespace _connectivity_bits;

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

     checkConcept<concepts::ReadWriteMap<typename Digraph::Node, int>,

       NodeMap>();

     typedef typename Digraph::Node Node;

     typedef typename Digraph::NodeIt NodeIt;

     typedef typename Digraph::Arc Arc;

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

       order.set(it, -1);

     }

     TopologicalSortVisitor<Digraph, NodeMap>

       visitor(order, countNodes(digraph));

     DfsVisit<Digraph, TopologicalSortVisitor<Digraph, NodeMap> >

       dfs(digraph, visitor);

     dfs.init();

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

       if (!dfs.reached(it)) {

         dfs.addSource(it);

         while (!dfs.emptyQueue()) {

            Arc arc = dfs.nextArc();

            Node target = digraph.target(arc);

            if (dfs.reached(target) && order[target] == -1) {

              return false;

            }

            dfs.processNextArc();

          }

       }

     }

     return true;

   }

   /// \ingroup graph_properties

   ///

   /// \brief Check whether an undirected graph is acyclic.

   ///

   /// This function checks whether the given undirected graph is acyclic.

   /// \return \c true if there is no cycle in the graph.

   /// \see dag()

   template <typename Graph>

   bool acyclic(const Graph& graph) {

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

     typedef typename Graph::Node Node;

     typedef typename Graph::NodeIt NodeIt;

     typedef typename Graph::Arc Arc;

     Dfs<Graph> dfs(graph);

     dfs.init();

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

       if (!dfs.reached(it)) {

         dfs.addSource(it);

         while (!dfs.emptyQueue()) {

           Arc arc = dfs.nextArc();

           Node source = graph.source(arc);

           Node target = graph.target(arc);

           if (dfs.reached(target) &&

               dfs.predArc(source) != graph.oppositeArc(arc)) {

             return false;

           }

           dfs.processNextArc();

         }

       }

     }

     return true;

   }

   /// \ingroup graph_properties

   ///

   /// \brief Check whether an undirected graph is tree.

   ///

   /// This function checks whether the given undirected graph is tree.

   /// \return \c true if the graph is acyclic and connected.

   /// \see acyclic(), connected()

   template <typename Graph>

   bool tree(const Graph& graph) {

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

     typedef typename Graph::Node Node;

     typedef typename Graph::NodeIt NodeIt;

     typedef typename Graph::Arc Arc;

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

     Dfs<Graph> dfs(graph);

     dfs.init();

     dfs.addSource(NodeIt(graph));

     while (!dfs.emptyQueue()) {

       Arc arc = dfs.nextArc();

       Node source = graph.source(arc);

       Node target = graph.target(arc);

       if (dfs.reached(target) &&

           dfs.predArc(source) != graph.oppositeArc(arc)) {

         return false;

       }

       dfs.processNextArc();

     }

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

       if (!dfs.reached(it)) {

         return false;

       }

     }

     return true;

   }

   namespace _connectivity_bits {

     template <typename Digraph>

     class BipartiteVisitor : public BfsVisitor<Digraph> {

     public:

       typedef typename Digraph::Arc Arc;

       typedef typename Digraph::Node Node;

       BipartiteVisitor(const Digraph& graph, bool& bipartite)

         : _graph(graph), _part(graph), _bipartite(bipartite) {}

       void start(const Node& node) {

         _part[node] = true;

       }

       void discover(const Arc& edge) {

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

       }

       void examine(const Arc& edge) {

         _bipartite = _bipartite &&

           _part[_graph.target(edge)] != _part[_graph.source(edge)];

       }

     private:

       const Digraph& _graph;

       typename Digraph::template NodeMap<bool> _part;

       bool& _bipartite;

     };

     template <typename Digraph, typename PartMap>

     class BipartitePartitionsVisitor : public BfsVisitor<Digraph> {

     public:

       typedef typename Digraph::Arc Arc;

       typedef typename Digraph::Node Node;

       BipartitePartitionsVisitor(const Digraph& graph,

                                  PartMap& part, bool& bipartite)

         : _graph(graph), _part(part), _bipartite(bipartite) {}

       void start(const Node& node) {

         _part.set(node, true);

       }

       void discover(const Arc& edge) {

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

       }

       void examine(const Arc& edge) {

         _bipartite = _bipartite &&

           _part[_graph.target(edge)] != _part[_graph.source(edge)];

       }

     private:

       const Digraph& _graph;

       PartMap& _part;

       bool& _bipartite;

     };

   }

   /// \ingroup graph_properties

   ///

   /// \brief Check whether an undirected graph is bipartite.

   ///

   /// The function checks whether the given undirected graph is bipartite.

   /// \return \c true if the graph is bipartite.

   ///

   /// \see bipartitePartitions()

   template<typename Graph>

   bool bipartite(const Graph &graph){

     using namespace _connectivity_bits;

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

     typedef typename Graph::NodeIt NodeIt;

     typedef typename Graph::ArcIt ArcIt;

     bool bipartite = true;

     BipartiteVisitor<Graph>

       visitor(graph, bipartite);

     BfsVisit<Graph, BipartiteVisitor<Graph> >

       bfs(graph, visitor);

     bfs.init();

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

       if(!bfs.reached(it)){

         bfs.addSource(it);

         while (!bfs.emptyQueue()) {

           bfs.processNextNode();

           if (!bipartite) return false;

         }

       }

     }

     return true;

   }

   /// \ingroup graph_properties

   ///

   /// \brief Find the bipartite partitions of an undirected graph.

   ///

   /// This function checks whether the given undirected graph is bipartite

   /// and gives back the bipartite partitions.

   ///

   /// \image html bipartite_partitions.png

   /// \image latex bipartite_partitions.eps "Bipartite partititions" width=\textwidth

   ///

   /// \param graph The undirected graph.

   /// \retval partMap A writable node map of \c bool (or convertible) value

   /// type. The values will be set to \c true for one component and

   /// \c false for the other one.

   /// \return \c true if the graph is bipartite, \c false otherwise.

   ///

   /// \see bipartite()

   template<typename Graph, typename NodeMap>

   bool bipartitePartitions(const Graph &graph, NodeMap &partMap){

     using namespace _connectivity_bits;

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

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

     typedef typename Graph::Node Node;

     typedef typename Graph::NodeIt NodeIt;

     typedef typename Graph::ArcIt ArcIt;

     bool bipartite = true;

     BipartitePartitionsVisitor<Graph, NodeMap>

       visitor(graph, partMap, bipartite);

     BfsVisit<Graph, BipartitePartitionsVisitor<Graph, NodeMap> >

       bfs(graph, visitor);

     bfs.init();

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

       if(!bfs.reached(it)){

         bfs.addSource(it);

         while (!bfs.emptyQueue()) {

           bfs.processNextNode();

           if (!bipartite) return false;

         }

       }

     }

     return true;

   }

   /// \ingroup graph_properties

   ///

   /// \brief Check whether the given graph contains no loop arcs/edges.

   ///

   /// This function returns \c true if there are no loop arcs/edges in

   /// the given graph. It works for both directed and undirected graphs.

   template <typename Graph>

   bool loopFree(const Graph& graph) {

     for (typename Graph::ArcIt it(graph); it != INVALID; ++it) {

       if (graph.source(it) == graph.target(it)) return false;

     }

     return true;

   }

   /// \ingroup graph_properties

   ///

   /// \brief Check whether the given graph contains no parallel arcs/edges.

   ///

   /// This function returns \c true if there are no parallel arcs/edges in

   /// the given graph. It works for both directed and undirected graphs.

   template <typename Graph>

   bool parallelFree(const Graph& graph) {

     typename Graph::template NodeMap<int> reached(graph, 0);

     int cnt = 1;

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

       for (typename Graph::OutArcIt a(graph, n); a != INVALID; ++a) {

         if (reached[graph.target(a)] == cnt) return false;

         reached[graph.target(a)] = cnt;

       }

       ++cnt;

     }

     return true;

   }

   /// \ingroup graph_properties

   ///

   /// \brief Check whether the given graph is simple.

   ///

   /// This function returns \c true if the given graph is simple, i.e.

   /// it contains no loop arcs/edges and no parallel arcs/edges.

   /// The function works for both directed and undirected graphs.

   /// \see loopFree(), parallelFree()

   template <typename Graph>

   bool simpleGraph(const Graph& graph) {

     typename Graph::template NodeMap<int> reached(graph, 0);

     int cnt = 1;

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

       reached[n] = cnt;

       for (typename Graph::OutArcIt a(graph, n); a != INVALID; ++a) {

         if (reached[graph.target(a)] == cnt) return false;

         reached[graph.target(a)] = cnt;

       }

       ++cnt;

     }

     return true;

   }

 } //namespace lemon

 #endif //LEMON_CONNECTIVITY_H