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

  *

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

  *

  * Copyright (C) 2003-2010

  * 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 concepts of graph components.

 #ifndef LEMON_CONCEPTS_GRAPH_COMPONENTS_H

 #define LEMON_CONCEPTS_GRAPH_COMPONENTS_H

 #include <lemon/core.h>

 #include <lemon/concepts/maps.h>

 #include <lemon/bits/alteration_notifier.h>

 namespace lemon {

   namespace concepts {

     /// \brief Concept class for \c Node, \c Arc and \c Edge types.

     ///

     /// This class describes the concept of \c Node, \c Arc and \c Edge

     /// subtypes of digraph and 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 that \c Node

     /// and \c Arc (or \c Edge) types should \e not derive from the same

     /// base class. For \c Node you should instantiate it with character

     /// \c 'n', for \c Arc with \c 'a' and for \c Edge with \c 'e'.

 #ifndef DOXYGEN

     template <char sel = '0'>

 #endif

     class GraphItem {

     public:

       /// \brief Default constructor.

       ///

       /// 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 Constructor for conversion from \c INVALID.

       ///

       /// Constructor for conversion from \c INVALID.

       /// It initializes the item to be invalid.

       /// \sa Invalid for more details.

       GraphItem(Invalid) {}

       /// \brief Assignment operator.

       ///

       /// Assignment operator for the item.

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

       /// \brief Assignment operator for INVALID.

       ///

       /// This operator makes the item invalid.

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

       /// \brief Equality operator.

       ///

       /// Equality operator.

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

       /// \brief Inequality operator.

       ///

       /// Inequality operator.

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

       /// \brief Ordering operator.

       ///

       /// This operator defines an ordering of the items.

       /// It makes possible to use graph item types as key types in

       /// associative containers (e.g. \c std::map).

       ///

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

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

       /// ordering of the items.

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

       template<typename _GraphItem>

       struct Constraints {

         void constraints() {

           _GraphItem i1;

           i1=INVALID;

           _GraphItem i2 = i1;

           _GraphItem i3 = INVALID;

           i1 = i2 = i3;

           bool b;

           ignore_unused_variable_warning(b);

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

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

           b = (ia < ib);

         }

         const _GraphItem &ia;

         const _GraphItem &ib;

         Constraints() {}

       };

     };

     /// \brief Base skeleton class for directed graphs.

     ///

     /// This class describes the base interface of directed graph types.

     /// All digraph %concepts have to conform to this class.

     /// It just provides types for nodes and arcs and functions

     /// to get the source and the target nodes of 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<'a'> Arc;

       /// \brief Return the source node of an arc.

       ///

       /// This function returns the source node of an arc.

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

       /// \brief Return the target node of an arc.

       ///

       /// This function returns the target node of an arc.

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

       /// \brief Return the opposite node on the given arc.

       ///

       /// This function returns 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;

         Constraints() {}

       };

     };

     /// \brief Base skeleton class for undirected graphs.

     ///

     /// This class describes the base interface of undirected graph types.

     /// All graph %concepts have to conform to this class.

     /// It extends the interface of \ref BaseDigraphComponent with an

     /// \c Edge type and functions to get the end nodes of edges,

     /// to convert from arcs to edges and to get both direction of edges.

     class BaseGraphComponent : public BaseDigraphComponent {

     public:

       typedef BaseGraphComponent Graph;

       typedef BaseDigraphComponent::Node Node;

       typedef BaseDigraphComponent::Arc Arc;

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

       ///

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

       /// Undirected graphs can be used as directed graphs, each edge is

       /// represented by two opposite directed arcs.

       class Edge : public GraphItem<'e'> {

         typedef GraphItem<'e'> Parent;

       public:

         /// \brief Default constructor.

         ///

         /// 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 Constructor for conversion from \c INVALID.

         ///

         /// Constructor for conversion from \c INVALID.

         /// It initializes the item to be invalid.

         /// \sa Invalid for more details.

         Edge(Invalid) {}

         /// \brief Constructor for conversion from an arc.

         ///

         /// Constructor for conversion from an arc.

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

         /// be convertible to the represented edge.

         Edge(const Arc&) {}

      };

       /// \brief Return one end node of an edge.

       ///

       /// This function returns one end node of an edge.

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

       /// \brief Return the other end node of an edge.

       ///

       /// This function returns the other end node of an edge.

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

       /// \brief Return a directed arc related to an edge.

       ///

       /// This function returns a directed arc from its direction and the

       /// represented edge.

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

       /// \brief Return a directed arc related to an edge.

       ///

       /// This function returns a directed arc from its source node and the

       /// represented edge.

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

       /// \brief Return 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 Return the opposite arc.

       ///

       /// This function returns the opposite arc, i.e. the arc representing

       /// the same edge and has opposite direction.

       Arc oppositeArc(const Arc&) 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<'e'>, 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, false);

             e = graph.direct(ue, n);

             e = graph.oppositeArc(e);

             ue = e;

             bool d = graph.direction(e);

             ignore_unused_variable_warning(d);

           }

         }

         const _Graph& graph;

       Constraints() {}

       };

     };

     /// \brief Base skeleton class for undirected bipartite graphs.

     ///

     /// This class describes the base interface of undirected

     /// bipartite graph types.  All bipartite graph %concepts have to

     /// conform to this class.  It extends the interface of \ref

     /// BaseGraphComponent with an \c Edge type and functions to get

     /// the end nodes of edges, to convert from arcs to edges and to

     /// get both direction of edges.

     class BaseBpGraphComponent : public BaseGraphComponent {

     public:

       typedef BaseBpGraphComponent BpGraph;

       typedef BaseDigraphComponent::Node Node;

       typedef BaseDigraphComponent::Arc Arc;

       /// \brief Class to represent red nodes.

       ///

       /// This class represents the red nodes of the graph. The red

       /// nodes can also be used as normal nodes.

       class RedNode : public Node {

         typedef Node Parent;

       public:

         /// \brief Default constructor.

         ///

         /// 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.

         RedNode() {}

         /// \brief Copy constructor.

         ///

         /// Copy constructor.

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

         /// \brief Constructor for conversion from \c INVALID.

         ///

         /// Constructor for conversion from \c INVALID.

         /// It initializes the item to be invalid.

         /// \sa Invalid for more details.

         RedNode(Invalid) {}

       };

       /// \brief Class to represent blue nodes.

       ///

       /// This class represents the blue nodes of the graph. The blue

       /// nodes can also be used as normal nodes.

       class BlueNode : public Node {

         typedef Node Parent;

       public:

         /// \brief Default constructor.

         ///

         /// 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.

         BlueNode() {}

         /// \brief Copy constructor.

         ///

         /// Copy constructor.

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

         /// \brief Constructor for conversion from \c INVALID.

         ///

         /// Constructor for conversion from \c INVALID.

         /// It initializes the item to be invalid.

         /// \sa Invalid for more details.

         BlueNode(Invalid) {}

         /// \brief Constructor for conversion from a node.

         ///

         /// Constructor for conversion from a node. The conversion can

         /// be invalid, since the Node can be member of the red

         /// set.

         BlueNode(const Node&) {}

       };

       /// \brief Gives back %true for red nodes.

       ///

       /// Gives back %true for red nodes.

       bool red(const Node&) const { return true; }

       /// \brief Gives back %true for blue nodes.

       ///

       /// Gives back %true for blue nodes.

       bool blue(const Node&) const { return true; }

       /// \brief Gives back the red end node of the edge.

       /// 

       /// Gives back the red end node of the edge.

       RedNode redNode(const Edge&) const { return RedNode(); }

       /// \brief Gives back the blue end node of the edge.

       /// 

       /// Gives back the blue end node of the edge.

       BlueNode blueNode(const Edge&) const { return BlueNode(); }

       /// \brief Converts the node to red node object.

       ///

       /// This function converts unsafely the node to red node

       /// object. It should be called only if the node is from the red

       /// partition or INVALID.

       RedNode asRedNodeUnsafe(const Node&) const { return RedNode(); }

       /// \brief Converts the node to blue node object.

       ///

       /// This function converts unsafely the node to blue node

       /// object. It should be called only if the node is from the red

       /// partition or INVALID.

       BlueNode asBlueNodeUnsafe(const Node&) const { return BlueNode(); }

       /// \brief Converts the node to red node object.

       ///

       /// This function converts safely the node to red node

       /// object. If the node is not from the red partition, then it

       /// returns INVALID.

       RedNode asRedNode(const Node&) const { return RedNode(); }

       /// \brief Converts the node to blue node object.

       ///

       /// This function converts unsafely the node to blue node

       /// object. If the node is not from the blue partition, then it

       /// returns INVALID.

       BlueNode asBlueNode(const Node&) const { return BlueNode(); }

       template <typename _BpGraph>

       struct Constraints {

         typedef typename _BpGraph::Node Node;

         typedef typename _BpGraph::RedNode RedNode;

         typedef typename _BpGraph::BlueNode BlueNode;

         typedef typename _BpGraph::Arc Arc;

         typedef typename _BpGraph::Edge Edge;

         void constraints() {

           checkConcept<BaseGraphComponent, _BpGraph>();

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

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

           {

             Node n;

             RedNode rn;

             BlueNode bn;

             Node rnan = rn;

             Node bnan = bn;

             Edge e;

             bool b;

             b = bpgraph.red(rnan);

             b = bpgraph.blue(bnan);

             rn = bpgraph.redNode(e);

             bn = bpgraph.blueNode(e);

             rn = bpgraph.asRedNodeUnsafe(rnan);

             bn = bpgraph.asBlueNodeUnsafe(bnan);

             rn = bpgraph.asRedNode(rnan);

             bn = bpgraph.asBlueNode(bnan);

             ignore_unused_variable_warning(b);

           }

         }

         const _BpGraph& bpgraph;

       };

     };

     /// \brief Skeleton class for \e idable directed graphs.

     ///

     /// This class describes the interface of \e idable directed graphs.

     /// It extends \ref BaseDigraphComponent with the core ID functions.

     /// The ids of the items must be unique and immutable.

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

     template <typename BAS = BaseDigraphComponent>

     class IDableDigraphComponent : public BAS {

     public:

       typedef BAS Base;

       typedef typename Base::Node Node;

       typedef typename Base::Arc Arc;

       /// \brief Return a unique integer id for the given node.

       ///

       /// This function returns a unique integer id for the given node.

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

       /// \brief Return the node by its unique id.

       ///

       /// This function returns the node by its unique id.

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

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

       Node nodeFromId(int) const { return INVALID; }

       /// \brief Return a unique integer id for the given arc.

       ///

       /// This function returns a unique integer id for the given arc.

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

       /// \brief Return the arc by its unique id.

       ///

       /// This function returns the arc by its unique id.

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

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

       Arc arcFromId(int) const { return INVALID; }

       /// \brief Return an integer greater or equal to the maximum

       /// node id.

       ///

       /// This function returns an integer greater or equal to the

       /// maximum node id.

       int maxNodeId() const { return -1; }

       /// \brief Return an integer greater or equal to the maximum

       /// arc id.

       ///

       /// This function returns 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;

           node=INVALID;

           int nid = digraph.id(node);

           nid = digraph.id(node);

           node = digraph.nodeFromId(nid);

           typename _Digraph::Arc arc;

           arc=INVALID;

           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;

         Constraints() {}

       };

     };

     /// \brief Skeleton class for \e idable undirected graphs.

     ///

     /// This class describes the interface of \e idable undirected

     /// graphs. It extends \ref IDableDigraphComponent with the core ID

     /// functions of undirected graphs.

     /// The ids of the items must be unique and immutable.

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

     template <typename BAS = BaseGraphComponent>

     class IDableGraphComponent : public IDableDigraphComponent<BAS> {

     public:

       typedef BAS Base;

       typedef typename Base::Edge Edge;

       using IDableDigraphComponent<Base>::id;

       /// \brief Return a unique integer id for the given edge.

       ///

       /// This function returns a unique integer id for the given edge.

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

       /// \brief Return the edge by its unique id.

       ///

       /// This function returns the edge by its unique id.

       /// If the graph does not contain an edge with the given id,

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

       Edge edgeFromId(int) const { return INVALID; }

       /// \brief Return an integer greater or equal to the maximum

       /// edge id.

       ///

       /// This function returns an integer greater or equal to the

       /// maximum edge id.

       int maxEdgeId() const { return -1; }

       template <typename _Graph>

       struct Constraints {

         void constraints() {

           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;

         Constraints() {}

       };

     };

     /// \brief Skeleton class for \e idable undirected bipartite graphs.

     ///

     /// This class describes the interface of \e idable undirected

     /// bipartite graphs. It extends \ref IDableGraphComponent with

     /// the core ID functions of undirected bipartite graphs. Beside

     /// the regular node ids, this class also provides ids within the

     /// the red and blue sets of the nodes. This concept is part of

     /// the BpGraph concept.

     template <typename BAS = BaseBpGraphComponent>

     class IDableBpGraphComponent : public IDableGraphComponent<BAS> {

     public:

       typedef BAS Base;

       typedef IDableGraphComponent<BAS> Parent;

       typedef typename Base::Node Node;

       typedef typename Base::RedNode RedNode;

       typedef typename Base::BlueNode BlueNode;

       using Parent::id;

       /// \brief Return a unique integer id for the given node in the red set.

       ///

       /// Return a unique integer id for the given node in the red set.

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

       /// \brief Return a unique integer id for the given node in the blue set.

       ///

       /// Return a unique integer id for the given node in the blue set.

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

       /// \brief Return an integer greater or equal to the maximum

       /// node id in the red set.

       ///

       /// Return an integer greater or equal to the maximum

       /// node id in the red set.

       int maxRedId() const { return -1; }

       /// \brief Return an integer greater or equal to the maximum

       /// node id in the blue set.

       ///

       /// Return an integer greater or equal to the maximum

       /// node id in the blue set.

       int maxBlueId() const { return -1; }

       template <typename _BpGraph>

       struct Constraints {

         void constraints() {

           checkConcept<IDableGraphComponent<Base>, _BpGraph>();

           typename _BpGraph::Node node;

           typename _BpGraph::RedNode red;

           typename _BpGraph::BlueNode blue;

           int rid = bpgraph.id(red);

           int bid = bpgraph.id(blue);

           rid = bpgraph.maxRedId();

           bid = bpgraph.maxBlueId();

           ignore_unused_variable_warning(rid);

           ignore_unused_variable_warning(bid);

         }

         const _BpGraph& bpgraph;

       };

     };

     /// \brief Concept class for \c NodeIt, \c ArcIt and \c EdgeIt types.

     ///

     /// This class describes the concept of \c NodeIt, \c ArcIt and

     /// \c EdgeIt subtypes of digraph and graph types.

     template <typename GR, typename Item>

     class GraphItemIt : public Item {

     public:

       /// \brief Default constructor.

       ///

       /// Default constructor.

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

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

       /// as uninitialized.

       GraphItemIt() {}

       /// \brief Copy constructor.

       ///

       /// Copy constructor.

       GraphItemIt(const GraphItemIt& it) : Item(it) {}

       /// \brief Constructor that sets the iterator to the first item.

       ///

       /// Constructor that sets the iterator to the first item.

       explicit GraphItemIt(const GR&) {}

       /// \brief Constructor for conversion from \c INVALID.

       ///

       /// Constructor for conversion from \c INVALID.

       /// It initializes the iterator to be invalid.

       /// \sa Invalid for more details.

       GraphItemIt(Invalid) {}

       /// \brief Assignment operator.

       ///

       /// Assignment operator for the iterator.

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

       /// \brief Increment the iterator.

       ///

       /// This operator increments the iterator, i.e. assigns it to the

       /// next item.

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

       /// \brief Equality operator

       ///

       /// 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

       ///

       /// Inequality 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;}

       template<typename _GraphItemIt>

       struct Constraints {

         void constraints() {

           checkConcept<GraphItem<>, _GraphItemIt>();

           _GraphItemIt it1(g);

           _GraphItemIt it2;

           _GraphItemIt it3 = it1;

           _GraphItemIt it4 = INVALID;

           ignore_unused_variable_warning(it3);

           ignore_unused_variable_warning(it4);

           it2 = ++it1;

           ++it2 = it1;

           ++(++it1);

           Item bi = it1;

           bi = it2;

         }

         const GR& g;

         Constraints() {}

       };

     };

     /// \brief Concept class for \c InArcIt, \c OutArcIt and

     /// \c IncEdgeIt types.

     ///

     /// This class describes the concept of \c InArcIt, \c OutArcIt

     /// and \c IncEdgeIt subtypes of digraph and graph types.

     ///

     /// \note Since these iterator classes do not inherit from the same

     /// base class, there is an additional template parameter (selector)

     /// \c sel. For \c InArcIt you should instantiate it with character

     /// \c 'i', for \c OutArcIt with \c 'o' and for \c IncEdgeIt with \c 'e'.

     template <typename GR,

               typename Item = typename GR::Arc,

               typename Base = typename GR::Node,

               char sel = '0'>

     class GraphIncIt : public Item {

     public:

       /// \brief Default constructor.

       ///

       /// Default constructor.

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

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

       /// as uninitialized.

       GraphIncIt() {}

       /// \brief Copy constructor.

       ///

       /// Copy constructor.

       GraphIncIt(const GraphIncIt& it) : Item(it) {}

       /// \brief Constructor that sets the iterator to the first

       /// incoming or outgoing arc.

       ///

       /// Constructor that sets the iterator to the first arc

       /// incoming to or outgoing from the given node.

       explicit GraphIncIt(const GR&, const Base&) {}

       /// \brief Constructor for conversion from \c INVALID.

       ///

       /// Constructor for conversion from \c INVALID.

       /// It initializes the iterator to be invalid.

       /// \sa Invalid for more details.

       GraphIncIt(Invalid) {}

       /// \brief Assignment operator.

       ///

       /// Assignment operator for the iterator.

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

       /// \brief Increment the iterator.

       ///

       /// This operator increments the iterator, i.e. assigns it to the

       /// next arc incoming to or outgoing from the given node.

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

       /// \brief Equality operator

       ///

       /// 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

       ///

       /// Inequality 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;}

       template <typename _GraphIncIt>

       struct Constraints {

         void constraints() {

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

           _GraphIncIt it1(graph, node);

           _GraphIncIt it2;

           _GraphIncIt it3 = it1;

           _GraphIncIt it4 = INVALID;

           ignore_unused_variable_warning(it3);

           ignore_unused_variable_warning(it4);

           it2 = ++it1;

           ++it2 = it1;

           ++(++it1);

           Item e = it1;

           e = it2;

         }

         const Base& node;

         const GR& graph;

         Constraints() {}

       };

     };

     /// \brief Skeleton class for iterable directed graphs.

     ///

     /// This class describes the interface of iterable directed

     /// graphs. It extends \ref BaseDigraphComponent with the core

     /// iterable interface.

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

     template <typename BAS = BaseDigraphComponent>

     class IterableDigraphComponent : public BAS {

     public:

       typedef BAS 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 Return the first node.

       ///

       /// This function gives back the first node in the iteration order.

       void first(Node&) const {}

       /// \brief Return the next node.

       ///

       /// This function gives back the next node in the iteration order.

       void next(Node&) const {}

       /// \brief Return the first arc.

       ///

       /// This function gives back the first arc in the iteration order.

       void first(Arc&) const {}

       /// \brief Return the next arc.

       ///

       /// This function gives back the next arc in the iteration order.

       void next(Arc&) const {}

       /// \brief Return the first arc incomming to the given node.

       ///

       /// This function gives back the first arc incomming to the

       /// given node.

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

       /// \brief Return the next arc incomming to the given node.

       ///

       /// This function gives back the next arc incomming to the

       /// given node.

       void nextIn(Arc&) const {}

       /// \brief Return the first arc outgoing form the given node.

       ///

       /// This function gives back the first arc outgoing form the

       /// given node.

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

       /// \brief Return the next arc outgoing form the given node.

       ///

       /// This function gives back the next arc outgoing form the

       /// given node.

       void nextOut(Arc&) const {}

       /// @}

       /// \name Class Based Iteration

       ///

       /// This interface provides iterator classes for 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 arc.

       ///

       /// This iterator goes through each arc.

       ///

       typedef GraphItemIt<Digraph, Arc> ArcIt;

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

       ///

       /// This iterator goes trough the \e incoming 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.

       ///

       /// This function gives back the base node of the iterator.

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

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

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

       ///

       /// This function gives back the running node of the iterator.

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

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

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

       ///

       /// This function gives back the base node of the iterator.

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

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

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

       ///

       /// This function gives back the running node of the iterator.

       /// It is always the target node 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;

             const typename _Digraph::InArcIt iait(INVALID);

             const typename _Digraph::OutArcIt oait(INVALID);

             n = digraph.baseNode(iait);

             n = digraph.runningNode(iait);

             n = digraph.baseNode(oait);

             n = digraph.runningNode(oait);

             ignore_unused_variable_warning(n);

           }

         }

         const _Digraph& digraph;

         Constraints() {}

       };

     };

     /// \brief Skeleton class for iterable undirected graphs.

     ///

     /// This class describes the interface of iterable undirected

     /// graphs. It extends \ref IterableDigraphComponent with the core

     /// iterable interface of undirected graphs.

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

     template <typename BAS = BaseGraphComponent>

     class IterableGraphComponent : public IterableDigraphComponent<BAS> {

     public:

       typedef BAS 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 edges.

       ///

       /// @{

       using IterableDigraphComponent<Base>::first;

       using IterableDigraphComponent<Base>::next;

       /// \brief Return the first edge.

       ///

       /// This function gives back the first edge in the iteration order.

       void first(Edge&) const {}

       /// \brief Return the next edge.

       ///

       /// This function gives back the next edge in the iteration order.

       void next(Edge&) const {}

       /// \brief Return the first edge incident to the given node.

       ///

       /// This function gives back the first edge incident to the given

       /// node. The bool parameter gives back the direction for which the

       /// source node of the directed arc representing the edge is the

       /// given node.

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

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

       /// given node.

       ///

       /// This function gives back the next edge incident to the given

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

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

       using IterableDigraphComponent<Base>::baseNode;

       using IterableDigraphComponent<Base>::runningNode;

       /// @}

       /// \name Class Based Iteration

       ///

       /// This interface provides iterator classes for edges.

       ///

       /// @{

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

       ///

       /// This iterator goes through each edge.

       typedef GraphItemIt<Graph, Edge> EdgeIt;

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

       /// node.

       ///

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

       /// node of a graph.

       typedef GraphIncIt<Graph, Edge, Node, 'e'> IncEdgeIt;

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

       ///

       /// This function gives back the base node of the iterator.

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

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

       ///

       /// This function gives back the running node of the iterator.

       Node runningNode(const IncEdgeIt&) 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, 'e'>, typename _Graph::IncEdgeIt>();

             typename _Graph::Node n;

             const typename _Graph::IncEdgeIt ieit(INVALID);

             n = graph.baseNode(ieit);

             n = graph.runningNode(ieit);

           }

         }

         const _Graph& graph;

         Constraints() {}

       };

     };

     /// \brief Skeleton class for iterable undirected bipartite graphs.

     ///

     /// This class describes the interface of iterable undirected

     /// bipartite graphs. It extends \ref IterableGraphComponent with

     /// the core iterable interface of undirected bipartite graphs.

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

     template <typename BAS = BaseBpGraphComponent>

     class IterableBpGraphComponent : public IterableGraphComponent<BAS> {

     public:

       typedef BAS Base;

       typedef typename Base::Node Node;

       typedef typename Base::RedNode RedNode;

       typedef typename Base::BlueNode BlueNode;

       typedef typename Base::Arc Arc;

       typedef typename Base::Edge Edge;

       typedef IterableBpGraphComponent BpGraph;

       using IterableGraphComponent<BAS>::first;

       using IterableGraphComponent<BAS>::next;

       /// \name Base Iteration

       ///

       /// This interface provides functions for iteration on red and blue nodes.

       ///

       /// @{

       /// \brief Return the first red node.

       ///

       /// This function gives back the first red node in the iteration order.

       void first(RedNode&) const {}

       /// \brief Return the next red node.

       ///

       /// This function gives back the next red node in the iteration order.

       void next(RedNode&) const {}

       /// \brief Return the first blue node.

       ///

       /// This function gives back the first blue node in the iteration order.

       void first(BlueNode&) const {}

       /// \brief Return the next blue node.

       ///

       /// This function gives back the next blue node in the iteration order.

       void next(BlueNode&) const {}

       /// @}

       /// \name Class Based Iteration

       ///

       /// This interface provides iterator classes for red and blue nodes.

       ///

       /// @{

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

       ///

       /// This iterator goes through each red node.

       typedef GraphItemIt<BpGraph, RedNode> RedNodeIt;

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

       ///

       /// This iterator goes through each blue node.

       typedef GraphItemIt<BpGraph, BlueNode> BlueNodeIt;

       /// @}

       template <typename _BpGraph>

       struct Constraints {

         void constraints() {

           checkConcept<IterableGraphComponent<Base>, _BpGraph>();

           typename _BpGraph::RedNode rn(INVALID);

           bpgraph.first(rn);

           bpgraph.next(rn); 

           typename _BpGraph::BlueNode bn(INVALID);

           bpgraph.first(bn);

           bpgraph.next(bn);

           checkConcept<GraphItemIt<_BpGraph, typename _BpGraph::RedNode>,

             typename _BpGraph::RedNodeIt>();

           checkConcept<GraphItemIt<_BpGraph, typename _BpGraph::BlueNode>,

             typename _BpGraph::BlueNodeIt>();

         }

         const _BpGraph& bpgraph;

       };

     };

     /// \brief Skeleton class for alterable directed graphs.

     ///

     /// This class describes the interface of alterable directed

     /// graphs. It extends \ref BaseDigraphComponent with the alteration

     /// notifier interface. It 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 be

     /// notified about it.

     template <typename BAS = BaseDigraphComponent>

     class AlterableDigraphComponent : public BAS {

     public:

       typedef BAS Base;

       typedef typename Base::Node Node;

       typedef typename Base::Arc Arc;

       /// Node alteration notifier class.

       typedef AlterationNotifier<AlterableDigraphComponent, Node>

       NodeNotifier;

       /// Arc alteration notifier class.

       typedef AlterationNotifier<AlterableDigraphComponent, Arc>

       ArcNotifier;

       mutable NodeNotifier node_notifier;

       mutable ArcNotifier arc_notifier;

       /// \brief Return the node alteration notifier.

       ///

       /// This function gives back the node alteration notifier.

       NodeNotifier& notifier(Node) const {

         return node_notifier;

       }

       /// \brief Return the arc alteration notifier.

       ///

       /// This function gives back the arc alteration notifier.

       ArcNotifier& notifier(Arc) const {

         return arc_notifier;

       }

       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;

         Constraints() {}

       };

     };

     /// \brief Skeleton class for alterable undirected graphs.

     ///

     /// This class describes the interface of alterable undirected

     /// graphs. It extends \ref AlterableDigraphComponent with the alteration

     /// notifier interface of undirected graphs. It implements

     /// an observer-notifier pattern for the edges. More

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

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

     /// notified about it.

     template <typename BAS = BaseGraphComponent>

     class AlterableGraphComponent : public AlterableDigraphComponent<BAS> {

     public:

       typedef BAS Base;

       typedef AlterableDigraphComponent<Base> Parent;

       typedef typename Base::Edge Edge;

       /// Edge alteration notifier class.

       typedef AlterationNotifier<AlterableGraphComponent, Edge>

       EdgeNotifier;

       mutable EdgeNotifier edge_notifier;

       using Parent::notifier;

       /// \brief Return the edge alteration notifier.

       ///

       /// This function gives back the edge alteration notifier.

       EdgeNotifier& notifier(Edge) const {

         return edge_notifier;

       }

       template <typename _Graph>

       struct Constraints {

         void constraints() {

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

           typename _Graph::EdgeNotifier& uen

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

           ignore_unused_variable_warning(uen);

         }

         const _Graph& graph;

         Constraints() {}

       };

     };

     /// \brief Skeleton class for alterable undirected bipartite graphs.

     ///

     /// This class describes the interface of alterable undirected

     /// bipartite graphs. It extends \ref AlterableGraphComponent with

     /// the alteration notifier interface of bipartite graphs. It

     /// implements an observer-notifier pattern for the red and blue

     /// nodes. More obsevers can be registered into the notifier and

     /// whenever an alteration occured in the graph all the observers

     /// will be notified about it.

     template <typename BAS = BaseBpGraphComponent>

     class AlterableBpGraphComponent : public AlterableGraphComponent<BAS> {

     public:

       typedef BAS Base;

       typedef AlterableGraphComponent<Base> Parent;

       typedef typename Base::RedNode RedNode;

       typedef typename Base::BlueNode BlueNode;

       /// Red node alteration notifier class.

       typedef AlterationNotifier<AlterableBpGraphComponent, RedNode>

       RedNodeNotifier;

       /// Blue node alteration notifier class.

       typedef AlterationNotifier<AlterableBpGraphComponent, BlueNode>

       BlueNodeNotifier;

       mutable RedNodeNotifier red_node_notifier;

       mutable BlueNodeNotifier blue_node_notifier;

       using Parent::notifier;

       /// \brief Return the red node alteration notifier.

       ///

       /// This function gives back the red node alteration notifier.

       RedNodeNotifier& notifier(RedNode) const {

         return red_node_notifier;

       }

       /// \brief Return the blue node alteration notifier.

       ///

       /// This function gives back the blue node alteration notifier.

       BlueNodeNotifier& notifier(BlueNode) const {

         return blue_node_notifier;

       }

       template <typename _BpGraph>

       struct Constraints {

         void constraints() {

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

           typename _BpGraph::RedNodeNotifier& rnn

             = bpgraph.notifier(typename _BpGraph::RedNode());

           typename _BpGraph::BlueNodeNotifier& bnn

             = bpgraph.notifier(typename _BpGraph::BlueNode());

           ignore_unused_variable_warning(rnn);

           ignore_unused_variable_warning(bnn);

         }

         const _BpGraph& bpgraph;

       };

     };

     /// \brief Concept class for standard graph maps.

     ///

     /// This class describes the concept of standard graph maps, i.e.

     /// the \c NodeMap, \c ArcMap and \c EdgeMap subtypes of digraph and

     /// graph types, which can be used for associating data to graph items.

     /// The standard graph maps must conform to the ReferenceMap concept.

     template <typename GR, typename K, typename V>

     class GraphMap : public ReferenceMap<K, V, V&, const V&> {

       typedef ReferenceMap<K, V, V&, const V&> Parent;

     public:

       /// The key type of the map.

       typedef K Key;

       /// The value type of the map.

       typedef V Value;

       /// The reference type of the map.

       typedef Value& Reference;

       /// The const reference type of the map.

       typedef const Value& ConstReference;

       // The reference map tag.

       typedef True ReferenceMapTag;

       /// \brief Construct a new map.

       ///

       /// Construct a new map for the graph.

       explicit GraphMap(const GR&) {}

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

       ///

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

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

     private:

       /// \brief Copy constructor.

       ///

       /// Copy Constructor.

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

       /// \brief Assignment operator.

       ///

       /// Assignment 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;

       }

     public:

       template<typename _Map>

       struct Constraints {

         void constraints() {

           checkConcept

             <ReferenceMap<Key, Value, Value&, const Value&>, _Map>();

           _Map m1(g);

           _Map m2(g,t);

           // Copy constructor

           // _Map m3(m);

           // Assignment operator

           // ReadMap<Key, Value> cmap;

           // m3 = cmap;

           ignore_unused_variable_warning(m1);

           ignore_unused_variable_warning(m2);

           // ignore_unused_variable_warning(m3);

         }

         const _Map &m;

         const GR &g;

         const typename GraphMap::Value &t;

         Constraints() {}

       };

     };

     /// \brief Skeleton class for mappable directed graphs.

     ///

     /// This class describes the interface of mappable directed graphs.

     /// It extends \ref BaseDigraphComponent with the standard digraph

     /// map classes, namely \c NodeMap and \c ArcMap.

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

     template <typename BAS = BaseDigraphComponent>

     class MappableDigraphComponent : public BAS  {

     public:

       typedef BAS Base;

       typedef typename Base::Node Node;

       typedef typename Base::Arc Arc;

       typedef MappableDigraphComponent Digraph;

       /// \brief Standard graph map for the nodes.

       ///

       /// Standard graph map for the nodes.

       /// It conforms to the ReferenceMap concept.

       template <typename V>

       class NodeMap : public GraphMap<MappableDigraphComponent, Node, V> {

         typedef GraphMap<MappableDigraphComponent, Node, V> Parent;

       public:

         /// \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 initalize the values.

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

           : Parent(digraph, value) {}

       private:

         /// \brief Copy constructor.

         ///

         /// Copy Constructor.

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

         /// \brief Assignment operator.

         ///

         /// Assignment operator.

         template <typename CMap>

         NodeMap& operator=(const CMap&) {

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

           return *this;

         }

       };

       /// \brief Standard graph map for the arcs.

       ///

       /// Standard graph map for the arcs.

       /// It conforms to the ReferenceMap concept.

       template <typename V>

       class ArcMap : public GraphMap<MappableDigraphComponent, Arc, V> {

         typedef GraphMap<MappableDigraphComponent, Arc, V> Parent;

       public:

         /// \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 initalize the values.

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

           : Parent(digraph, value) {}

       private:

         /// \brief Copy constructor.

         ///

         /// Copy Constructor.

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

         /// \brief Assignment operator.

         ///

         /// Assignment operator.

         template <typename CMap>

         ArcMap& operator=(const CMap&) {

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

           return *this;

         }

       };

       template <typename _Digraph>

       struct Constraints {

         struct Dummy {

           int value;

           Dummy() : value(0) {}

           Dummy(int _v) : value(_v) {}

         };

         void constraints() {

           checkConcept<Base, _Digraph>();

           { // int map test

             typedef typename _Digraph::template NodeMap<int> IntNodeMap;

             checkConcept<GraphMap<_Digraph, typename _Digraph::Node, int>,

               IntNodeMap >();

           } { // bool map test

             typedef typename _Digraph::template NodeMap<bool> BoolNodeMap;

             checkConcept<GraphMap<_Digraph, typename _Digraph::Node, bool>,

               BoolNodeMap >();

           } { // Dummy map test

             typedef typename _Digraph::template NodeMap<Dummy> DummyNodeMap;

             checkConcept<GraphMap<_Digraph, typename _Digraph::Node, Dummy>,

               DummyNodeMap >();

           }

           { // int map test

             typedef typename _Digraph::template ArcMap<int> IntArcMap;

             checkConcept<GraphMap<_Digraph, typename _Digraph::Arc, int>,

               IntArcMap >();

           } { // bool map test

             typedef typename _Digraph::template ArcMap<bool> BoolArcMap;

             checkConcept<GraphMap<_Digraph, typename _Digraph::Arc, bool>,

               BoolArcMap >();

           } { // Dummy map test

             typedef typename _Digraph::template ArcMap<Dummy> DummyArcMap;

             checkConcept<GraphMap<_Digraph, typename _Digraph::Arc, Dummy>,

               DummyArcMap >();

           }

         }

         const _Digraph& digraph;

         Constraints() {}

       };

     };

     /// \brief Skeleton class for mappable undirected graphs.

     ///

     /// This class describes the interface of mappable undirected graphs.

     /// It extends \ref MappableDigraphComponent with the standard graph

     /// map class for edges (\c EdgeMap).

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

     template <typename BAS = BaseGraphComponent>

     class MappableGraphComponent : public MappableDigraphComponent<BAS>  {

     public:

       typedef BAS Base;

       typedef typename Base::Edge Edge;

       typedef MappableGraphComponent Graph;

       /// \brief Standard graph map for the edges.

       ///

       /// Standard graph map for the edges.

       /// It conforms to the ReferenceMap concept.

       template <typename V>

       class EdgeMap : public GraphMap<MappableGraphComponent, Edge, V> {

         typedef GraphMap<MappableGraphComponent, Edge, V> Parent;

       public:

         /// \brief Construct a new map.

         ///

         /// Construct a new map for the graph.

         explicit EdgeMap(const MappableGraphComponent& graph)

           : Parent(graph) {}

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

         ///

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

         EdgeMap(const MappableGraphComponent& graph, const V& value)

           : Parent(graph, value) {}

       private:

         /// \brief Copy constructor.

         ///

         /// Copy Constructor.

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

         /// \brief Assignment operator.

         ///

         /// Assignment operator.

         template <typename CMap>

         EdgeMap& operator=(const CMap&) {

           checkConcept<ReadMap<Edge, V>, CMap>();

           return *this;

         }

       };

       template <typename _Graph>

       struct Constraints {

         struct Dummy {

           int value;

           Dummy() : value(0) {}

           Dummy(int _v) : value(_v) {}

         };

         void constraints() {

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

           { // int map test

             typedef typename _Graph::template EdgeMap<int> IntEdgeMap;

             checkConcept<GraphMap<_Graph, typename _Graph::Edge, int>,

               IntEdgeMap >();

           } { // bool map test

             typedef typename _Graph::template EdgeMap<bool> BoolEdgeMap;

             checkConcept<GraphMap<_Graph, typename _Graph::Edge, bool>,

               BoolEdgeMap >();

           } { // Dummy map test

             typedef typename _Graph::template EdgeMap<Dummy> DummyEdgeMap;

             checkConcept<GraphMap<_Graph, typename _Graph::Edge, Dummy>,

               DummyEdgeMap >();

           }

         }

         const _Graph& graph;

         Constraints() {}

       };

     };

     /// \brief Skeleton class for mappable undirected bipartite graphs.

     ///

     /// This class describes the interface of mappable undirected

     /// bipartite graphs.  It extends \ref MappableGraphComponent with

     /// the standard graph map class for red and blue nodes (\c

     /// RedNodeMap and BlueNodeMap). This concept is part of the

     /// BpGraph concept.

     template <typename BAS = BaseBpGraphComponent>

     class MappableBpGraphComponent : public MappableGraphComponent<BAS>  {

     public:

       typedef BAS Base;

       typedef typename Base::Node Node;

       typedef MappableBpGraphComponent BpGraph;

       /// \brief Standard graph map for the red nodes.

       ///

       /// Standard graph map for the red nodes.

       /// It conforms to the ReferenceMap concept.

       template <typename V>

       class RedNodeMap : public GraphMap<MappableBpGraphComponent, Node, V> {

         typedef GraphMap<MappableBpGraphComponent, Node, V> Parent;

       public:

         /// \brief Construct a new map.

         ///

         /// Construct a new map for the graph.

         explicit RedNodeMap(const MappableBpGraphComponent& graph)

           : Parent(graph) {}

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

         ///

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

         RedNodeMap(const MappableBpGraphComponent& graph, const V& value)

           : Parent(graph, value) {}

       private:

         /// \brief Copy constructor.

         ///

         /// Copy Constructor.

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

         /// \brief Assignment operator.

         ///

         /// Assignment operator.

         template <typename CMap>

         RedNodeMap& operator=(const CMap&) {

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

           return *this;

         }

       };

       /// \brief Standard graph map for the blue nodes.

       ///

       /// Standard graph map for the blue nodes.

       /// It conforms to the ReferenceMap concept.

       template <typename V>

       class BlueNodeMap : public GraphMap<MappableBpGraphComponent, Node, V> {

         typedef GraphMap<MappableBpGraphComponent, Node, V> Parent;

       public:

         /// \brief Construct a new map.

         ///

         /// Construct a new map for the graph.

         explicit BlueNodeMap(const MappableBpGraphComponent& graph)

           : Parent(graph) {}

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

         ///

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

         BlueNodeMap(const MappableBpGraphComponent& graph, const V& value)

           : Parent(graph, value) {}

       private:

         /// \brief Copy constructor.

         ///

         /// Copy Constructor.

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

         /// \brief Assignment operator.

         ///

         /// Assignment operator.

         template <typename CMap>

         BlueNodeMap& operator=(const CMap&) {

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

           return *this;

         }

       };

       template <typename _BpGraph>

       struct Constraints {

         struct Dummy {

           int value;

           Dummy() : value(0) {}

           Dummy(int _v) : value(_v) {}

         };

         void constraints() {

           checkConcept<MappableGraphComponent<Base>, _BpGraph>();

           { // int map test

             typedef typename _BpGraph::template RedNodeMap<int>

               IntRedNodeMap;

             checkConcept<GraphMap<_BpGraph, typename _BpGraph::RedNode, int>,

               IntRedNodeMap >();

           } { // bool map test

             typedef typename _BpGraph::template RedNodeMap<bool>

               BoolRedNodeMap;

             checkConcept<GraphMap<_BpGraph, typename _BpGraph::RedNode, bool>,

               BoolRedNodeMap >();

           } { // Dummy map test

             typedef typename _BpGraph::template RedNodeMap<Dummy>

               DummyRedNodeMap;

             checkConcept<GraphMap<_BpGraph, typename _BpGraph::RedNode, Dummy>,

               DummyRedNodeMap >();

           }

           { // int map test

             typedef typename _BpGraph::template BlueNodeMap<int>

               IntBlueNodeMap;

             checkConcept<GraphMap<_BpGraph, typename _BpGraph::BlueNode, int>,

               IntBlueNodeMap >();

           } { // bool map test

             typedef typename _BpGraph::template BlueNodeMap<bool>

               BoolBlueNodeMap;

             checkConcept<GraphMap<_BpGraph, typename _BpGraph::BlueNode, bool>,

               BoolBlueNodeMap >();

           } { // Dummy map test

             typedef typename _BpGraph::template BlueNodeMap<Dummy>

               DummyBlueNodeMap;

             checkConcept<GraphMap<_BpGraph, typename _BpGraph::BlueNode, Dummy>,

               DummyBlueNodeMap >();

           }

         }

         const _BpGraph& bpgraph;

       };

     };

     /// \brief Skeleton class for extendable directed graphs.

     ///

     /// This class describes the interface of extendable directed graphs.

     /// It extends \ref BaseDigraphComponent with functions for adding

     /// nodes and arcs to the digraph.

     /// This concept requires \ref AlterableDigraphComponent.

     template <typename BAS = BaseDigraphComponent>

     class ExtendableDigraphComponent : public BAS {

     public:

       typedef BAS Base;

       typedef typename Base::Node Node;

       typedef typename Base::Arc Arc;

       /// \brief Add a new node to the digraph.

       ///

       /// This function adds a new node to the digraph.

       Node addNode() {

         return INVALID;

       }

       /// \brief Add a new arc connecting the given two nodes.

       ///

       /// This function adds a new arc connecting the given two nodes

       /// of the digraph.

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

         return INVALID;

       }

       template <typename _Digraph>

       struct Constraints {

         void constraints() {

           checkConcept<Base, _Digraph>();

           typename _Digraph::Node node_a, node_b;

           node_a = digraph.addNode();

           node_b = digraph.addNode();

           typename _Digraph::Arc arc;

           arc = digraph.addArc(node_a, node_b);

         }

         _Digraph& digraph;

         Constraints() {}

       };

     };

     /// \brief Skeleton class for extendable undirected graphs.

     ///

     /// This class describes the interface of extendable undirected graphs.

     /// It extends \ref BaseGraphComponent with functions for adding

     /// nodes and edges to the graph.

     /// This concept requires \ref AlterableGraphComponent.

     template <typename BAS = BaseGraphComponent>

     class ExtendableGraphComponent : public BAS {

     public:

       typedef BAS Base;

       typedef typename Base::Node Node;

       typedef typename Base::Edge Edge;

       /// \brief Add a new node to the digraph.

       ///

       /// This function adds a new node to the digraph.

       Node addNode() {

         return INVALID;

       }

       /// \brief Add a new edge connecting the given two nodes.

       ///

       /// This function adds a new edge connecting the given two nodes

       /// of the graph.

       Edge addEdge(const Node&, const Node&) {

         return INVALID;

       }

       template <typename _Graph>

       struct Constraints {

         void constraints() {

           checkConcept<Base, _Graph>();

           typename _Graph::Node node_a, node_b;

           node_a = graph.addNode();

           node_b = graph.addNode();

           typename _Graph::Edge edge;

           edge = graph.addEdge(node_a, node_b);

         }

         _Graph& graph;

         Constraints() {}

       };

     };

     /// \brief Skeleton class for extendable undirected bipartite graphs.

     ///

     /// This class describes the interface of extendable undirected

     /// bipartite graphs. It extends \ref BaseGraphComponent with

     /// functions for adding nodes and edges to the graph. This

     /// concept requires \ref AlterableBpGraphComponent.

     template <typename BAS = BaseBpGraphComponent>

     class ExtendableBpGraphComponent : public BAS {

     public:

       typedef BAS Base;

       typedef typename Base::Node Node;

       typedef typename Base::RedNode RedNode;

       typedef typename Base::BlueNode BlueNode;

       typedef typename Base::Edge Edge;

       /// \brief Add a new red node to the digraph.

       ///

       /// This function adds a red new node to the digraph.

       RedNode addRedNode() {

         return INVALID;

       }

       /// \brief Add a new blue node to the digraph.

       ///

       /// This function adds a blue new node to the digraph.

       BlueNode addBlueNode() {

         return INVALID;

       }

       /// \brief Add a new edge connecting the given two nodes.

       ///

       /// This function adds a new edge connecting the given two nodes

       /// of the graph. The first node has to be a red node, and the

       /// second one a blue node.

       Edge addEdge(const RedNode&, const BlueNode&) {

         return INVALID;

       }

       Edge addEdge(const BlueNode&, const RedNode&) {

         return INVALID;

       }

       template <typename _BpGraph>

       struct Constraints {

         void constraints() {

           checkConcept<Base, _BpGraph>();

           typename _BpGraph::RedNode red_node;

           typename _BpGraph::BlueNode blue_node;

           red_node = bpgraph.addRedNode();

           blue_node = bpgraph.addBlueNode();

           typename _BpGraph::Edge edge;

           edge = bpgraph.addEdge(red_node, blue_node);

           edge = bpgraph.addEdge(blue_node, red_node);

         }

         _BpGraph& bpgraph;

       };

     };

     /// \brief Skeleton class for erasable directed graphs.

     ///

     /// This class describes the interface of erasable directed graphs.

     /// It extends \ref BaseDigraphComponent with functions for removing

     /// nodes and arcs from the digraph.

     /// This concept requires \ref AlterableDigraphComponent.

     template <typename BAS = BaseDigraphComponent>

     class ErasableDigraphComponent : public BAS {

     public:

       typedef BAS Base;

       typedef typename Base::Node Node;

       typedef typename Base::Arc Arc;

       /// \brief Erase a node from the digraph.

       ///

       /// This function erases the given node from the digraph and all arcs

       /// connected to the node.

       void erase(const Node&) {}

       /// \brief Erase an arc from the digraph.

       ///

       /// This function erases the given arc from the digraph.

       void erase(const Arc&) {}

       template <typename _Digraph>

       struct Constraints {

         void constraints() {

           checkConcept<Base, _Digraph>();

           const typename _Digraph::Node node(INVALID);

           digraph.erase(node);

           const typename _Digraph::Arc arc(INVALID);

           digraph.erase(arc);

         }

         _Digraph& digraph;

         Constraints() {}

       };

     };

     /// \brief Skeleton class for erasable undirected graphs.

     ///

     /// This class describes the interface of erasable undirected graphs.

     /// It extends \ref BaseGraphComponent with functions for removing

     /// nodes and edges from the graph.

     /// This concept requires \ref AlterableGraphComponent.

     template <typename BAS = BaseGraphComponent>

     class ErasableGraphComponent : public BAS {

     public:

       typedef BAS Base;

       typedef typename Base::Node Node;

       typedef typename Base::Edge Edge;

       /// \brief Erase a node from the graph.

       ///

       /// This function erases the given node from the graph and all edges

       /// connected to the node.

       void erase(const Node&) {}

       /// \brief Erase an edge from the digraph.

       ///

       /// This function erases the given edge from the digraph.

       void erase(const Edge&) {}

       template <typename _Graph>

       struct Constraints {

         void constraints() {

           checkConcept<Base, _Graph>();

           const typename _Graph::Node node(INVALID);

           graph.erase(node);

           const typename _Graph::Edge edge(INVALID);

           graph.erase(edge);

         }

         _Graph& graph;

         Constraints() {}

       };

     };

     /// \brief Skeleton class for erasable undirected graphs.

     ///

     /// This class describes the interface of erasable undirected

     /// bipartite graphs. It extends \ref BaseBpGraphComponent with

     /// functions for removing nodes and edges from the graph. This

     /// concept requires \ref AlterableBpGraphComponent.

     template <typename BAS = BaseBpGraphComponent>

     class ErasableBpGraphComponent : public ErasableGraphComponent<BAS> {};

     /// \brief Skeleton class for clearable directed graphs.

     ///

     /// This class describes the interface of clearable directed graphs.

     /// It extends \ref BaseDigraphComponent with a function for clearing

     /// the digraph.

     /// This concept requires \ref AlterableDigraphComponent.

     template <typename BAS = BaseDigraphComponent>

     class ClearableDigraphComponent : public BAS {

     public:

       typedef BAS Base;

       /// \brief Erase all nodes and arcs from the digraph.

       ///

       /// This function erases all nodes and arcs from the digraph.

       void clear() {}

       template <typename _Digraph>

       struct Constraints {

         void constraints() {

           checkConcept<Base, _Digraph>();

           digraph.clear();

         }

         _Digraph& digraph;

         Constraints() {}

       };

     };

     /// \brief Skeleton class for clearable undirected graphs.

     ///

     /// This class describes the interface of clearable undirected graphs.

     /// It extends \ref BaseGraphComponent with a function for clearing

     /// the graph.

     /// This concept requires \ref AlterableGraphComponent.

     template <typename BAS = BaseGraphComponent>

     class ClearableGraphComponent : public ClearableDigraphComponent<BAS> {};

     /// \brief Skeleton class for clearable undirected biparite graphs.

     ///

     /// This class describes the interface of clearable undirected

     /// bipartite graphs. It extends \ref BaseBpGraphComponent with a

     /// function for clearing the graph.  This concept requires \ref

     /// AlterableBpGraphComponent.

     template <typename BAS = BaseBpGraphComponent>

     class ClearableBpGraphComponent : public ClearableGraphComponent<BAS> {};

   }

 }

 #endif