/* -*- 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;

          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 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 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;

          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;

          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 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;

      /// \brief Return the node alteration notifier.

      ///

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

      NodeNotifier& notifier(Node) const {

         return NodeNotifier();

      }

      /// \brief Return the arc alteration notifier.

      ///

      /// This function 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;

        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 typename Base::Edge Edge;

      /// Edge alteration notifier class.

      typedef AlterationNotifier<AlterableGraphComponent, Edge>

      EdgeNotifier;

      /// \brief Return the edge alteration notifier.

      ///

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

      EdgeNotifier& notifier(Edge) const {

        return EdgeNotifier();

      }

      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);