/* -*- C++ -*-

  *

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

  *

  * Copyright (C) 2003-2007

  * 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.

  *

  */

 ///\ingroup graph_concepts

 ///\file

 ///\brief The concept of graph components.

 #ifndef LEMON_CONCEPT_GRAPH_COMPONENTS_H

 #define LEMON_CONCEPT_GRAPH_COMPONENTS_H

 #include <lemon/bits/invalid.h>

 #include <lemon/concepts/maps.h>

 #include <lemon/bits/alteration_notifier.h>

 namespace lemon {

   namespace concepts {

     /// \brief Skeleton class for graph Node and Arc types

     ///

     /// This class describes the interface of Node and Arc (and Edge

     /// in undirected graphs) subtypes of graph types.

     ///

     /// \note This class is a template class so that we can use it to

     /// create graph skeleton classes. The reason for this is than Node

     /// and Arc types should \em not derive from the same base class.

     /// For Node you should instantiate it with character 'n' and for Arc

     /// with 'a'.

 #ifndef DOXYGEN

     template <char _selector = '0'>

 #endif

     class GraphItem {

     public:

       /// \brief Default constructor.

       ///      

       /// \warning The default constructor is not required to set

       /// the item to some well-defined value. So you should consider it

       /// as uninitialized.

       GraphItem() {}

       /// \brief Copy constructor.

       ///

       /// Copy constructor.

       ///

       GraphItem(const GraphItem &) {}

       /// \brief Invalid constructor \& conversion.

       ///

       /// This constructor initializes the item to be invalid.

       /// \sa Invalid for more details.

       GraphItem(Invalid) {}

       /// \brief Assign operator for nodes.

       ///

       /// The nodes are assignable. 

       ///

       GraphItem& operator=(GraphItem const&) { return *this; }

       /// \brief Equality operator.

       ///

       /// Two iterators are equal if and only if they represents the

       /// same node in the graph or both are invalid.

       bool operator==(GraphItem) const { return false; }

       /// \brief Inequality operator.

       ///

       /// \sa operator==(const Node& n)

       ///

       bool operator!=(GraphItem) const { return false; }

       /// \brief Artificial ordering operator.

       ///

       /// To allow the use of graph descriptors as key type in std::map or

       /// similar associative container we require this.

       ///

       /// \note This operator only have to define some strict ordering of

       /// the items; this order has nothing to do with the iteration

       /// ordering of the items.

       bool operator<(GraphItem) const { return false; }

       template<typename _GraphItem>

       struct Constraints {

 	void constraints() {

 	  _GraphItem i1;

 	  _GraphItem i2 = i1;

 	  _GraphItem i3 = INVALID;

 	  i1 = i2 = i3;

 	  bool b;

 	  //	  b = (ia == ib) && (ia != ib) && (ia < ib);

 	  b = (ia == ib) && (ia != ib);

 	  b = (ia == INVALID) && (ib != INVALID);

           b = (ia < ib);

 	}

 	const _GraphItem &ia;

 	const _GraphItem &ib;

       };

     };

     /// \brief An empty base directed graph class.

     ///  

     /// This class provides the minimal set of features needed for a

     /// directed graph structure. All digraph concepts have to be

     /// conform to this base directed graph. It just provides types

     /// for nodes and arcs and functions to get the source and the

     /// target of the arcs.

     class BaseDigraphComponent {

     public:

       typedef BaseDigraphComponent Digraph;

       /// \brief Node class of the digraph.

       ///

       /// This class represents the Nodes of the digraph. 

       ///

       typedef GraphItem<'n'> Node;

       /// \brief Arc class of the digraph.

       ///

       /// This class represents the Arcs of the digraph. 

       ///

       typedef GraphItem<'e'> Arc;

       /// \brief Gives back the target node of an arc.

       ///

       /// Gives back the target node of an arc.

       ///

       Node target(const Arc&) const { return INVALID;}

       /// \brief Gives back the source node of an arc.

       ///

       /// Gives back the source node of an arc.

       ///

       Node source(const Arc&) const { return INVALID;}

       /// \brief Gives back the opposite node on the given arc.

       ///

       /// Gives back the opposite node on the given arc.

       Node oppositeNode(const Node&, const Arc&) const {

         return INVALID;

       }

       template <typename _Digraph>

       struct Constraints {

 	typedef typename _Digraph::Node Node;

 	typedef typename _Digraph::Arc Arc;

 	void constraints() {

 	  checkConcept<GraphItem<'n'>, Node>();

 	  checkConcept<GraphItem<'a'>, Arc>();

 	  {

 	    Node n;

 	    Arc e(INVALID);

 	    n = digraph.source(e);

 	    n = digraph.target(e);

             n = digraph.oppositeNode(n, e);

 	  }      

 	}

 	const _Digraph& digraph;

       };

     };

     /// \brief An empty base undirected graph class.

     ///  

     /// This class provides the minimal set of features needed for an

     /// undirected graph structure. All undirected graph concepts have

     /// to be conform to this base graph. It just provides types for

     /// nodes, arcs and edges and functions to get the

     /// source and the target of the arcs and edges,

     /// conversion from arcs to edges and function to get

     /// both direction of the edges.

     class BaseGraphComponent : public BaseDigraphComponent {

     public:

       typedef BaseDigraphComponent::Node Node;

       typedef BaseDigraphComponent::Arc Arc;

       /// \brief Undirected arc class of the graph.

       ///

       /// This class represents the edges of the graph.

       /// The undirected graphs can be used as a directed graph which

       /// for each arc contains the opposite arc too so the graph is

       /// bidirected. The edge represents two opposite

       /// directed arcs.

       class Edge : public GraphItem<'u'> {

       public:

         typedef GraphItem<'u'> Parent;

         /// \brief Default constructor.

         ///      

         /// \warning The default constructor is not required to set

         /// the item to some well-defined value. So you should consider it

         /// as uninitialized.

         Edge() {}

         /// \brief Copy constructor.

         ///

         /// Copy constructor.

         ///

         Edge(const Edge &) : Parent() {}

         /// \brief Invalid constructor \& conversion.

         ///

         /// This constructor initializes the item to be invalid.

         /// \sa Invalid for more details.

         Edge(Invalid) {}

         /// \brief Converter from arc to edge.

         ///

         /// Besides the core graph item functionality each arc should

         /// be convertible to the represented edge. 

         Edge(const Arc&) {}

         /// \brief Assign arc to edge.

         ///

         /// Besides the core graph item functionality each arc should

         /// be convertible to the represented edge. 

         Edge& operator=(const Arc&) { return *this; }

       };

       /// \brief Returns the direction of the arc.

       ///

       /// Returns the direction of the arc. Each arc represents an

       /// edge with a direction. It gives back the

       /// direction.

       bool direction(const Arc&) const { return true; }

       /// \brief Returns the directed arc.

       ///

       /// Returns the directed arc from its direction and the

       /// represented edge.

       Arc direct(const Edge&, bool) const { return INVALID;} 

       /// \brief Returns the directed arc.

       ///

       /// Returns the directed arc from its source and the

       /// represented edge.

       Arc direct(const Edge&, const Node&) const { return INVALID;} 

       /// \brief Returns the opposite arc.

       ///

       /// Returns the opposite arc. It is the arc representing the

       /// same edge and has opposite direction.

       Arc oppositeArc(const Arc&) const { return INVALID;}

       /// \brief Gives back one ending of an edge.

       ///

       /// Gives back one ending of an edge.

       Node u(const Edge&) const { return INVALID;}

       /// \brief Gives back the other ending of an edge.

       ///

       /// Gives back the other ending of an edge.

       Node v(const Edge&) const { return INVALID;}

       template <typename _Graph>

       struct Constraints {

 	typedef typename _Graph::Node Node;

 	typedef typename _Graph::Arc Arc;

 	typedef typename _Graph::Edge Edge;

 	void constraints() {

           checkConcept<BaseDigraphComponent, _Graph>();

 	  checkConcept<GraphItem<'u'>, Edge>();

 	  {

 	    Node n;

 	    Edge ue(INVALID);

             Arc e;

 	    n = graph.u(ue);

 	    n = graph.v(ue);

             e = graph.direct(ue, true);

             e = graph.direct(ue, n);

             e = graph.oppositeArc(e);

             ue = e;

             bool d = graph.direction(e);

             ignore_unused_variable_warning(d);

 	  }      

 	}

 	const _Graph& graph;

       };

     };

     /// \brief An empty idable base digraph class.

     ///  

     /// This class provides beside the core digraph features

     /// core id functions for the digraph structure.

     /// The most of the base digraphs should be conform to this concept.

     /// The id's are unique and immutable.

     template <typename _Base = BaseDigraphComponent>

     class IDableDigraphComponent : public _Base {

     public:

       typedef _Base Base;

       typedef typename Base::Node Node;

       typedef typename Base::Arc Arc;

       /// \brief Gives back an unique integer id for the Node. 

       ///

       /// Gives back an unique integer id for the Node. 

       ///

       int id(const Node&) const { return -1;}

       /// \brief Gives back the node by the unique id.

       ///

       /// Gives back the node by the unique id.

       /// If the digraph does not contain node with the given id

       /// then the result of the function is undetermined. 

       Node nodeFromId(int) const { return INVALID;}

       /// \brief Gives back an unique integer id for the Arc. 

       ///

       /// Gives back an unique integer id for the Arc. 

       ///

       int id(const Arc&) const { return -1;}

       /// \brief Gives back the arc by the unique id.

       ///

       /// Gives back the arc by the unique id.

       /// If the digraph does not contain arc with the given id

       /// then the result of the function is undetermined. 

       Arc arcFromId(int) const { return INVALID;}

       /// \brief Gives back an integer greater or equal to the maximum

       /// Node id.

       ///

       /// Gives back an integer greater or equal to the maximum Node

       /// id.

       int maxNodeId() const { return -1;}

       /// \brief Gives back an integer greater or equal to the maximum

       /// Arc id.

       ///

       /// Gives back an integer greater or equal to the maximum Arc

       /// id.

       int maxArcId() const { return -1;}

       template <typename _Digraph>

       struct Constraints {

 	void constraints() {

 	  checkConcept<Base, _Digraph >();

 	  typename _Digraph::Node node;

 	  int nid = digraph.id(node);

 	  nid = digraph.id(node);

 	  node = digraph.nodeFromId(nid);

 	  typename _Digraph::Arc arc;

 	  int eid = digraph.id(arc);

 	  eid = digraph.id(arc);

 	  arc = digraph.arcFromId(eid);

 	  nid = digraph.maxNodeId();

 	  ignore_unused_variable_warning(nid);

 	  eid = digraph.maxArcId();

 	  ignore_unused_variable_warning(eid);

 	}

 	const _Digraph& digraph;

       };

     };

     /// \brief An empty idable base undirected graph class.

     ///  

     /// This class provides beside the core undirected graph features

     /// core id functions for the undirected graph structure.  The

     /// most of the base undirected graphs should be conform to this

     /// concept.  The id's are unique and immutable.

     template <typename _Base = BaseGraphComponent>

     class IDableGraphComponent : public IDableDigraphComponent<_Base> {

     public:

       typedef _Base Base;

       typedef typename Base::Edge Edge;

       using IDableDigraphComponent<_Base>::id;

       /// \brief Gives back an unique integer id for the Edge. 

       ///

       /// Gives back an unique integer id for the Edge. 

       ///

       int id(const Edge&) const { return -1;}

       /// \brief Gives back the edge by the unique id.

       ///

       /// Gives back the edge by the unique id.  If the

       /// graph does not contain arc with the given id then the

       /// result of the function is undetermined.

       Edge edgeFromId(int) const { return INVALID;}

       /// \brief Gives back an integer greater or equal to the maximum

       /// Edge id.

       ///

       /// Gives back an integer greater or equal to the maximum Edge

       /// id.

       int maxEdgeId() const { return -1;}

       template <typename _Graph>

       struct Constraints {

 	void constraints() {

 	  checkConcept<Base, _Graph >();

 	  checkConcept<IDableDigraphComponent<Base>, _Graph >();

 	  typename _Graph::Edge edge;

 	  int ueid = graph.id(edge);

 	  ueid = graph.id(edge);

 	  edge = graph.edgeFromId(ueid);

 	  ueid = graph.maxEdgeId();

 	  ignore_unused_variable_warning(ueid);

 	}

 	const _Graph& graph;

       };

     };

     /// \brief Skeleton class for graph NodeIt and ArcIt

     ///

     /// Skeleton class for graph NodeIt and ArcIt.

     ///

     template <typename _Graph, typename _Item>

     class GraphItemIt : public _Item {

     public:

       /// \brief Default constructor.

       ///

       /// @warning The default constructor sets the iterator

       /// to an undefined value.

       GraphItemIt() {}

       /// \brief Copy constructor.

       ///

       /// Copy constructor.

       ///

       GraphItemIt(const GraphItemIt& ) {}

       /// \brief Sets the iterator to the first item.

       ///

       /// Sets the iterator to the first item of \c the graph.

       ///

       explicit GraphItemIt(const _Graph&) {}

       /// \brief Invalid constructor \& conversion.

       ///

       /// This constructor initializes the item to be invalid.

       /// \sa Invalid for more details.

       GraphItemIt(Invalid) {}

       /// \brief Assign operator for items.

       ///

       /// The items are assignable. 

       ///

       GraphItemIt& operator=(const GraphItemIt&) { return *this; }      

       /// \brief Next item.

       /// 

       /// Assign the iterator to the next item.

       ///

       GraphItemIt& operator++() { return *this; }

       /// \brief Equality operator

       /// 

       /// Two iterators are equal if and only if they point to the

       /// same object or both are invalid.

       bool operator==(const GraphItemIt&) const { return true;}

       /// \brief Inequality operator

       ///	

       /// \sa operator==(Node n)

       ///

       bool operator!=(const GraphItemIt&) const { return true;}

       template<typename _GraphItemIt>

       struct Constraints {

 	void constraints() {

 	  _GraphItemIt it1(g);	

 	  _GraphItemIt it2;

 	  it2 = ++it1;

 	  ++it2 = it1;

 	  ++(++it1);

 	  _Item bi = it1;

 	  bi = it2;

 	}

 	_Graph& g;

       };

     };

     /// \brief Skeleton class for graph InArcIt and OutArcIt

     ///

     /// \note Because InArcIt and OutArcIt may not inherit from the same

     /// base class, the _selector is a additional template parameter. For 

     /// InArcIt you should instantiate it with character 'i' and for 

     /// OutArcIt with 'o'.

     template <typename _Graph,

 	      typename _Item = typename _Graph::Arc,

               typename _Base = typename _Graph::Node, 

 	      char _selector = '0'>

     class GraphIncIt : public _Item {

     public:

       /// \brief Default constructor.

       ///

       /// @warning The default constructor sets the iterator

       /// to an undefined value.

       GraphIncIt() {}

       /// \brief Copy constructor.

       ///

       /// Copy constructor.

       ///

       GraphIncIt(GraphIncIt const& gi) : _Item(gi) {}

       /// \brief Sets the iterator to the first arc incoming into or outgoing 

       /// from the node.

       ///

       /// Sets the iterator to the first arc incoming into or outgoing 

       /// from the node.

       ///

       explicit GraphIncIt(const _Graph&, const _Base&) {}

       /// \brief Invalid constructor \& conversion.

       ///

       /// This constructor initializes the item to be invalid.

       /// \sa Invalid for more details.

       GraphIncIt(Invalid) {}

       /// \brief Assign operator for iterators.

       ///

       /// The iterators are assignable. 

       ///

       GraphIncIt& operator=(GraphIncIt const&) { return *this; }      

       /// \brief Next item.

       ///

       /// Assign the iterator to the next item.

       ///

       GraphIncIt& operator++() { return *this; }

       /// \brief Equality operator

       ///

       /// Two iterators are equal if and only if they point to the

       /// same object or both are invalid.

       bool operator==(const GraphIncIt&) const { return true;}

       /// \brief Inequality operator

       ///

       /// \sa operator==(Node n)

       ///

       bool operator!=(const GraphIncIt&) const { return true;}

       template <typename _GraphIncIt>

       struct Constraints {

 	void constraints() {

 	  checkConcept<GraphItem<_selector>, _GraphIncIt>();

 	  _GraphIncIt it1(graph, node);

 	  _GraphIncIt it2;

 	  it2 = ++it1;

 	  ++it2 = it1;

 	  ++(++it1);

 	  _Item e = it1;

 	  e = it2;

 	}

 	_Item arc;

 	_Base node;

 	_Graph graph;

 	_GraphIncIt it;

       };

     };

     /// \brief An empty iterable digraph class.

     ///

     /// This class provides beside the core digraph features

     /// iterator based iterable interface for the digraph structure.

     /// This concept is part of the Digraph concept.

     template <typename _Base = BaseDigraphComponent>

     class IterableDigraphComponent : public _Base {

     public:

       typedef _Base Base;

       typedef typename Base::Node Node;

       typedef typename Base::Arc Arc;

       typedef IterableDigraphComponent Digraph;

       /// \name Base iteration

       /// 

       /// This interface provides functions for iteration on digraph items

       ///

       /// @{  

       /// \brief Gives back the first node in the iterating order.

       ///      

       /// Gives back the first node in the iterating order.

       ///     

       void first(Node&) const {}

       /// \brief Gives back the next node in the iterating order.

       ///

       /// Gives back the next node in the iterating order.

       ///     

       void next(Node&) const {}

       /// \brief Gives back the first arc in the iterating order.

       ///

       /// Gives back the first arc in the iterating order.

       ///     

       void first(Arc&) const {}

       /// \brief Gives back the next arc in the iterating order.

       ///

       /// Gives back the next arc in the iterating order.

       ///     

       void next(Arc&) const {}

       /// \brief Gives back the first of the arcs point to the given

       /// node.

       ///

       /// Gives back the first of the arcs point to the given node.

       ///     

       void firstIn(Arc&, const Node&) const {}

       /// \brief Gives back the next of the arcs points to the given

       /// node.

       ///

       /// Gives back the next of the arcs points to the given node.

       ///

       void nextIn(Arc&) const {}

       /// \brief Gives back the first of the arcs start from the

       /// given node.

       ///      

       /// Gives back the first of the arcs start from the given node.

       ///     

       void firstOut(Arc&, const Node&) const {}

       /// \brief Gives back the next of the arcs start from the given

       /// node.

       ///

       /// Gives back the next of the arcs start from the given node.

       ///     

       void nextOut(Arc&) const {}

       /// @}

       /// \name Class based iteration

       /// 

       /// This interface provides functions for iteration on digraph items

       ///

       /// @{

       /// \brief This iterator goes through each node.

       ///

       /// This iterator goes through each node.

       ///

       typedef GraphItemIt<Digraph, Node> NodeIt;

       /// \brief This iterator goes through each node.

       ///

       /// This iterator goes through each node.

       ///

       typedef GraphItemIt<Digraph, Arc> ArcIt;

       /// \brief This iterator goes trough the incoming arcs of a node.

       ///

       /// This iterator goes trough the \e inccoming arcs of a certain node

       /// of a digraph.

       typedef GraphIncIt<Digraph, Arc, Node, 'i'> InArcIt;

       /// \brief This iterator goes trough the outgoing arcs of a node.

       ///

       /// This iterator goes trough the \e outgoing arcs of a certain node

       /// of a digraph.

       typedef GraphIncIt<Digraph, Arc, Node, 'o'> OutArcIt;

       /// \brief The base node of the iterator.

       ///

       /// Gives back the base node of the iterator.

       /// It is always the target of the pointed arc.

       Node baseNode(const InArcIt&) const { return INVALID; }

       /// \brief The running node of the iterator.

       ///

       /// Gives back the running node of the iterator.

       /// It is always the source of the pointed arc.

       Node runningNode(const InArcIt&) const { return INVALID; }

       /// \brief The base node of the iterator.

       ///

       /// Gives back the base node of the iterator.

       /// It is always the source of the pointed arc.

       Node baseNode(const OutArcIt&) const { return INVALID; }

       /// \brief The running node of the iterator.

       ///

       /// Gives back the running node of the iterator.

       /// It is always the target of the pointed arc.

       Node runningNode(const OutArcIt&) const { return INVALID; }

       /// @}

       template <typename _Digraph> 

       struct Constraints {

 	void constraints() {

 	  checkConcept<Base, _Digraph>();

           {

             typename _Digraph::Node node(INVALID);      

             typename _Digraph::Arc arc(INVALID);

             {

               digraph.first(node);

               digraph.next(node);

             }

             {

               digraph.first(arc);

               digraph.next(arc);

             }

             {

               digraph.firstIn(arc, node);

               digraph.nextIn(arc);

             }

             {

               digraph.firstOut(arc, node);

               digraph.nextOut(arc);

             }

           }           

           {

             checkConcept<GraphItemIt<_Digraph, typename _Digraph::Arc>,

               typename _Digraph::ArcIt >();

             checkConcept<GraphItemIt<_Digraph, typename _Digraph::Node>,

               typename _Digraph::NodeIt >();

             checkConcept<GraphIncIt<_Digraph, typename _Digraph::Arc, 

               typename _Digraph::Node, 'i'>, typename _Digraph::InArcIt>();

             checkConcept<GraphIncIt<_Digraph, typename _Digraph::Arc, 

               typename _Digraph::Node, 'o'>, typename _Digraph::OutArcIt>();

             typename _Digraph::Node n;

             typename _Digraph::InArcIt ieit(INVALID);

             typename _Digraph::OutArcIt oeit(INVALID);

             n = digraph.baseNode(ieit);

             n = digraph.runningNode(ieit);

             n = digraph.baseNode(oeit);

             n = digraph.runningNode(oeit);

             ignore_unused_variable_warning(n);

           }

         }

 	const _Digraph& digraph;

       };

     };

     /// \brief An empty iterable undirected graph class.

     ///

     /// This class provides beside the core graph features iterator

     /// based iterable interface for the undirected graph structure.

     /// This concept is part of the Graph concept.

     template <typename _Base = BaseGraphComponent>

     class IterableGraphComponent : public IterableDigraphComponent<_Base> {

     public:

       typedef _Base Base;

       typedef typename Base::Node Node;

       typedef typename Base::Arc Arc;

       typedef typename Base::Edge Edge;

       typedef IterableGraphComponent Graph;

       /// \name Base iteration

       /// 

       /// This interface provides functions for iteration on graph items

       /// @{  

       using IterableDigraphComponent<_Base>::first;

       using IterableDigraphComponent<_Base>::next;

       /// \brief Gives back the first edge in the iterating

       /// order.

       ///

       /// Gives back the first edge in the iterating order.

       ///     

       void first(Edge&) const {}

       /// \brief Gives back the next edge in the iterating

       /// order.

       ///

       /// Gives back the next edge in the iterating order.

       ///     

       void next(Edge&) const {}

       /// \brief Gives back the first of the edges from the

       /// given node.

       ///

       /// Gives back the first of the edges from the given

       /// node. The bool parameter gives back that direction which

       /// gives a good direction of the edge so the source of the

       /// directed arc is the given node.

       void firstInc(Edge&, bool&, const Node&) const {}

       /// \brief Gives back the next of the edges from the

       /// given node.

       ///

       /// Gives back the next of the edges from the given

       /// node. The bool parameter should be used as the \c firstInc()

       /// use it.

       void nextInc(Edge&, bool&) const {}

       using IterableDigraphComponent<_Base>::baseNode;

       using IterableDigraphComponent<_Base>::runningNode;

       /// @}

       /// \name Class based iteration

       /// 

       /// This interface provides functions for iteration on graph items

       ///

       /// @{

       /// \brief This iterator goes through each node.

       ///

       /// This iterator goes through each node.

       typedef GraphItemIt<Graph, Edge> EdgeIt;

       /// \brief This iterator goes trough the incident arcs of a

       /// node.

       ///

       /// This iterator goes trough the incident arcs of a certain

       /// node of a graph.

       typedef GraphIncIt<Graph, Edge, Node, 'u'> IncArcIt;

       /// \brief The base node of the iterator.

       ///

       /// Gives back the base node of the iterator.

       Node baseNode(const IncArcIt&) const { return INVALID; }

       /// \brief The running node of the iterator.

       ///

       /// Gives back the running node of the iterator.

       Node runningNode(const IncArcIt&) const { return INVALID; }

       /// @}

       template <typename _Graph> 

       struct Constraints {

 	void constraints() {

 	  checkConcept<IterableDigraphComponent<Base>, _Graph>();

           {

             typename _Graph::Node node(INVALID);

             typename _Graph::Edge edge(INVALID);

             bool dir;

             {

               graph.first(edge);

               graph.next(edge);

             }

             {

               graph.firstInc(edge, dir, node);

               graph.nextInc(edge, dir);

             }

           }	

           {

             checkConcept<GraphItemIt<_Graph, typename _Graph::Edge>,

               typename _Graph::EdgeIt >();

             checkConcept<GraphIncIt<_Graph, typename _Graph::Edge, 

               typename _Graph::Node, 'u'>, typename _Graph::IncArcIt>();

             typename _Graph::Node n;

             typename _Graph::IncArcIt ueit(INVALID);

             n = graph.baseNode(ueit);

             n = graph.runningNode(ueit);

           }

         }

 	const _Graph& graph;

       };

     };

     /// \brief An empty alteration notifier digraph class.

     ///  

     /// This class provides beside the core digraph features alteration

     /// notifier interface for the digraph structure.  This implements

     /// an observer-notifier pattern for each digraph item. More

     /// obsevers can be registered into the notifier and whenever an

     /// alteration occured in the digraph all the observers will

     /// notified about it.

     template <typename _Base = BaseDigraphComponent>

     class AlterableDigraphComponent : public _Base {

     public:

       typedef _Base Base;

       typedef typename Base::Node Node;

       typedef typename Base::Arc Arc;

       /// The node observer registry.

       typedef AlterationNotifier<AlterableDigraphComponent, Node> 

       NodeNotifier;

       /// The arc observer registry.

       typedef AlterationNotifier<AlterableDigraphComponent, Arc> 

       ArcNotifier;

       /// \brief Gives back the node alteration notifier.

       ///

       /// Gives back the node alteration notifier.

       NodeNotifier& notifier(Node) const {

 	return NodeNotifier();

       }

       /// \brief Gives back the arc alteration notifier.

       ///

       /// Gives back the arc alteration notifier.

       ArcNotifier& notifier(Arc) const {

 	return ArcNotifier();

       }

       template <typename _Digraph> 

       struct Constraints {

 	void constraints() {

 	  checkConcept<Base, _Digraph>();

           typename _Digraph::NodeNotifier& nn 

             = digraph.notifier(typename _Digraph::Node());

           typename _Digraph::ArcNotifier& en 

             = digraph.notifier(typename _Digraph::Arc());

           ignore_unused_variable_warning(nn);

           ignore_unused_variable_warning(en);

 	}

 	const _Digraph& digraph;

       };

     };

     /// \brief An empty alteration notifier undirected graph class.

     ///  

     /// This class provides beside the core graph features alteration

     /// notifier interface for the graph structure.  This implements

     /// an observer-notifier pattern for each graph item. More

     /// obsevers can be registered into the notifier and whenever an

     /// alteration occured in the graph all the observers will

     /// notified about it.

     template <typename _Base = BaseGraphComponent>

     class AlterableGraphComponent : public AlterableDigraphComponent<_Base> {

     public:

       typedef _Base Base;

       typedef typename Base::Edge Edge;

       /// The arc observer registry.

       typedef AlterationNotifier<AlterableGraphComponent, Edge> 

       EdgeNotifier;

       /// \brief Gives back the arc alteration notifier.

       ///

       /// Gives back the arc alteration notifier.

       EdgeNotifier& notifier(Edge) const {

 	return EdgeNotifier();

       }

       template <typename _Graph> 

       struct Constraints {

 	void constraints() {

 	  checkConcept<AlterableGraphComponent<Base>, _Graph>();

           typename _Graph::EdgeNotifier& uen 

             = graph.notifier(typename _Graph::Edge());

           ignore_unused_variable_warning(uen);

 	}

 	const _Graph& graph;

       };

     };

     /// \brief Class describing the concept of graph maps

     /// 

     /// This class describes the common interface of the graph maps

     /// (NodeMap, ArcMap), that is \ref maps-page "maps" which can be used to

     /// associate data to graph descriptors (nodes or arcs).

     template <typename _Graph, typename _Item, typename _Value>

     class GraphMap : public ReadWriteMap<_Item, _Value> {

     public:

       typedef ReadWriteMap<_Item, _Value> Parent;

       /// The graph type of the map.

       typedef _Graph Graph;

       /// The key type of the map.

       typedef _Item Key;

       /// The value type of the map.

       typedef _Value Value;

       /// \brief Construct a new map.

       ///

       /// Construct a new map for the graph.

       explicit GraphMap(const Graph&) {}

       /// \brief Construct a new map with default value.

       ///

       /// Construct a new map for the graph and initalise the values.

       GraphMap(const Graph&, const Value&) {}

       /// \brief Copy constructor.

       ///

       /// Copy Constructor.

       GraphMap(const GraphMap&) : Parent() {}

       /// \brief Assign operator.

       ///

       /// Assign operator. It does not mofify the underlying graph,

       /// it just iterates on the current item set and set the  map

       /// with the value returned by the assigned map. 

       template <typename CMap>

       GraphMap& operator=(const CMap&) { 

         checkConcept<ReadMap<Key, Value>, CMap>();

         return *this;

       }

       template<typename _Map>

       struct Constraints {

 	void constraints() {

 	  checkConcept<ReadWriteMap<Key, Value>, _Map >();

 	  // Construction with a graph parameter

 	  _Map a(g);

 	  // Constructor with a graph and a default value parameter

 	  _Map a2(g,t);

 	  // Copy constructor.

 	  _Map b(c);

           ReadMap<Key, Value> cmap;

           b = cmap;

 	  ignore_unused_variable_warning(a2);

 	  ignore_unused_variable_warning(b);

 	}

 	const _Map &c;

 	const Graph &g;

 	const typename GraphMap::Value &t;

       };

     };

     /// \brief An empty mappable digraph class.

     ///

     /// This class provides beside the core digraph features

     /// map interface for the digraph structure.

     /// This concept is part of the Digraph concept.

     template <typename _Base = BaseDigraphComponent>

     class MappableDigraphComponent : public _Base  {

     public:

       typedef _Base Base;

       typedef typename Base::Node Node;

       typedef typename Base::Arc Arc;

       typedef MappableDigraphComponent Digraph;

       /// \brief ReadWrite map of the nodes.

       ///

       /// ReadWrite map of the nodes.

       ///

       template <typename _Value>

       class NodeMap : public GraphMap<Digraph, Node, _Value> {

       public:

         typedef GraphMap<MappableDigraphComponent, Node, _Value> Parent;

 	/// \brief Construct a new map.

 	///

 	/// Construct a new map for the digraph.

 	explicit NodeMap(const MappableDigraphComponent& digraph) 

           : Parent(digraph) {}

 	/// \brief Construct a new map with default value.

 	///

 	/// Construct a new map for the digraph and initalise the values.

 	NodeMap(const MappableDigraphComponent& digraph, const _Value& value)

           : Parent(digraph, value) {}

 	/// \brief Copy constructor.

 	///

 	/// Copy Constructor.

 	NodeMap(const NodeMap& nm) : Parent(nm) {}

 	/// \brief Assign operator.

 	///

 	/// Assign operator.

         template <typename CMap>

         NodeMap& operator=(const CMap&) { 

           checkConcept<ReadMap<Node, _Value>, CMap>();

           return *this;

         }

       };

       /// \brief ReadWrite map of the arcs.

       ///

       /// ReadWrite map of the arcs.

       ///

       template <typename _Value>

       class ArcMap : public GraphMap<Digraph, Arc, _Value> {

       public:

         typedef GraphMap<MappableDigraphComponent, Arc, _Value> Parent;

 	/// \brief Construct a new map.

 	///

 	/// Construct a new map for the digraph.

 	explicit ArcMap(const MappableDigraphComponent& digraph) 

           : Parent(digraph) {}

 	/// \brief Construct a new map with default value.

 	///

 	/// Construct a new map for the digraph and initalise the values.

 	ArcMap(const MappableDigraphComponent& digraph, const _Value& value)

           : Parent(digraph, value) {}

 	/// \brief Copy constructor.

 	///

 	/// Copy Constructor.

 	ArcMap(const ArcMap& nm) : Parent(nm) {}

 	/// \brief Assign operator.

 	///

 	/// Assign operator.

         template <typename CMap>

         ArcMap& operator=(const CMap&) { 

           checkConcept<ReadMap<Arc, _Value>, CMap>();

           return *this;

         }

       };