/* -*- C++ -*-

 *

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

 *

 * Copyright (C) 2003-2008

 * Egervary Jeno Kombinatorikus Optimalizalasi Kutatocsoport

 * (Egervary Research Group on Combinatorial Optimization, EGRES).

 *

 * Permission to use, modify and distribute this software is granted

 * provided that this copyright notice appears in all copies. For

 * precise terms see the accompanying LICENSE file.

 *

 * This software is provided "AS IS" with no warranty of any kind,

 * express or implied, and with no claim as to its suitability for any

 * purpose.

 *

 */

///\ingroup graph_concepts

///\file

///\brief The concept of graph components.

#ifndef LEMON_CONCEPT_GRAPH_COMPONENTS_H

#define LEMON_CONCEPT_GRAPH_COMPONENTS_H

#include <lemon/bits/invalid.h>

#include <lemon/concepts/maps.h>

#include <lemon/bits/alteration_notifier.h>

namespace lemon {

  namespace concepts {

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

    ///

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

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

    ///

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

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

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

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

    /// with 'e'.

#ifndef DOXYGEN

    template <char _selector = '0'>

#endif

    class GraphItem {

    public:

      /// \brief Default constructor.

      ///      

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

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

      /// as uninitialized.

      GraphItem() {}

      /// \brief Copy constructor.

      ///

      /// Copy constructor.

      ///

      GraphItem(const GraphItem &) {}

      /// \brief Invalid constructor \& conversion.

      ///

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

      /// \sa Invalid for more details.

      GraphItem(Invalid) {}

      /// \brief Assign operator for nodes.

      ///

      /// The nodes are assignable. 

      ///

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

      /// \brief Equality operator.

      ///

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

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

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

      /// \brief Inequality operator.

      ///

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

      ///

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

      /// \brief Artificial ordering operator.

      ///

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

      /// similar associative container we require this.

      ///

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

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

      /// ordering of the items.

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

      template<typename _GraphItem>

      struct Constraints {

	void constraints() {

	  _GraphItem i1;

	  _GraphItem i2 = i1;

	  _GraphItem i3 = INVALID;

	  i1 = i2 = i3;

	  bool b;

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

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

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

          b = (ia < ib);

	}

	const _GraphItem &ia;

	const _GraphItem &ib;

      };

    };

    /// \brief An empty base graph class.

    ///  

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

    /// structure. All graph concepts have to be conform to this base

    /// graph. It just provides types for nodes and edges and functions to

    /// get the source and the target of the edges.

    class BaseGraphComponent {

    public:

      typedef BaseGraphComponent Graph;

      /// \brief Node class of the graph.

      ///

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

      ///

      typedef GraphItem<'n'> Node;

      /// \brief Edge class of the graph.

      ///

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

      ///

      typedef GraphItem<'e'> Edge;

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

      ///

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

      ///

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

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

      ///

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

      ///

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

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

      ///

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

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

        return INVALID;

      }

      template <typename _Graph>

      struct Constraints {

	typedef typename _Graph::Node Node;

	typedef typename _Graph::Edge Edge;

	void constraints() {

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

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

	  {

	    Node n;

	    Edge e(INVALID);

	    n = graph.source(e);

	    n = graph.target(e);

            n = graph.oppositeNode(n, e);

	  }      

	}

	const _Graph& graph;

      };

    };

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

    ///  

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

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

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

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

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

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

    /// both direction of the undirected edges.

    class BaseUGraphComponent : public BaseGraphComponent {

    public:

      typedef BaseGraphComponent::Node Node;

      typedef BaseGraphComponent::Edge Edge;

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

      ///

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

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

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

      /// bidirected. The undirected edge represents two opposite

      /// directed edges.

      class UEdge : public GraphItem<'u'> {

      public:

        typedef GraphItem<'u'> Parent;

        /// \brief Default constructor.

        ///      

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

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

        /// as uninitialized.

        UEdge() {}

        /// \brief Copy constructor.

        ///

        /// Copy constructor.

        ///

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

        /// \brief Invalid constructor \& conversion.

        ///

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

        /// \sa Invalid for more details.

        UEdge(Invalid) {}

        /// \brief Converter from edge to undirected edge.

        ///

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

        /// be convertible to the represented undirected edge. 

        UEdge(const Edge&) {}

        /// \brief Assign edge to undirected edge.

        ///

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

        /// be convertible to the represented undirected edge. 

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

      };

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

      ///

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

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

      /// direction.

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

      /// \brief Returns the directed edge.

      ///

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

      /// represented undirected edge.

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

      /// \brief Returns the directed edge.

      ///

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

      /// represented undirected edge.

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

      /// \brief Returns the opposite edge.

      ///

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

      /// same undirected edge and has opposite direction.

      Edge oppositeEdge(const Edge&) const { return INVALID;}

      /// \brief Gives back the target node of an undirected edge.

      ///

      /// Gives back the target node of an undirected edge. The name

      /// target is a little confusing because the undirected edge

      /// does not have target but it just means that one of the end 

      /// node.

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

      /// \brief Gives back the source node of an undirected edge.

      ///

      /// Gives back the source node of an undirected edge. The name

      /// source is a little confusing because the undirected edge

      /// does not have source but it just means that one of the end 

      /// node.

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

      template <typename _Graph>

      struct Constraints {

	typedef typename _Graph::Node Node;

	typedef typename _Graph::Edge Edge;

	typedef typename _Graph::UEdge UEdge;

	void constraints() {

          checkConcept<BaseGraphComponent, _Graph>();

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

	  {

	    Node n;

	    UEdge ue(INVALID);

            Edge e;

	    n = graph.source(ue);

	    n = graph.target(ue);

            e = graph.direct(ue, true);

            e = graph.direct(ue, n);

            e = graph.oppositeEdge(e);

            ue = e;

            bool d = graph.direction(e);

            ignore_unused_variable_warning(d);

	  }      

	}

	const _Graph& graph;

      };

    };

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

    ///  

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

    /// bipartite undirected graph structure. All bipartite undirected

    /// graph concepts have to be conform to this base graph. It just

    /// provides types for nodes, A-nodes, B-nodes, edges and

    /// undirected edges and functions to get the source and the

    /// target of the edges and undirected edges, conversion from

    /// edges to undirected edges and function to get both direction

    /// of the undirected edges.

    class BaseBpUGraphComponent : public BaseUGraphComponent {

    public:

      typedef BaseUGraphComponent::Node Node;

      typedef BaseUGraphComponent::Edge Edge;

      typedef BaseUGraphComponent::UEdge UEdge;

      /// \brief Helper class for A-nodes.

      ///

      /// This class is just a helper class for A-nodes, it is not

      /// suggested to use it directly. It can be converted easily to

      /// node and vice versa. The usage of this class is limited

      /// to use just as template parameters for special map types. 

      class ANode : public Node {

      public:

        typedef Node Parent;

        /// \brief Default constructor.

        ///      

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

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

        /// as uninitialized.

        ANode() {}

        /// \brief Copy constructor.

        ///

        /// Copy constructor.

        ///

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

        /// \brief Invalid constructor \& conversion.

        ///

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

        /// \sa Invalid for more details.

        ANode(Invalid) {}

        /// \brief Converter from node to A-node.

        ///

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

        /// be convertible to the represented A-node if it is it possible. 

        ANode(const Node&) {}

        /// \brief Assign node to A-node.

        ///

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

        /// be convertible to the represented A-node if it is it possible. 

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

      };

      /// \brief Helper class for B-nodes.

      ///

      /// This class is just a helper class for B-nodes, it is not

      /// suggested to use it directly. It can be converted easily to

      /// node and vice versa. The usage of this class is limited

      /// to use just as template parameters for special map types. 

      class BNode : public Node {

      public:

        typedef Node Parent;

        /// \brief Default constructor.

        ///      

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

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

        /// as uninitialized.

        BNode() {}

        /// \brief Copy constructor.

        ///

        /// Copy constructor.

        ///

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

        /// \brief Invalid constructor \& conversion.

        ///

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

        /// \sa Invalid for more details.

        BNode(Invalid) {}

        /// \brief Converter from node to B-node.

        ///

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

        /// be convertible to the represented B-node if it is it possible. 

        BNode(const Node&) {}

        /// \brief Assign node to B-node.

        ///

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

        /// be convertible to the represented B-node if it is it possible. 

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

      };

      /// \brief Gives back %true when the node is A-node.

      ///

      /// Gives back %true when the node is A-node.

      bool aNode(const Node&) const { return false; }

      /// \brief Gives back %true when the node is B-node.

      ///

      /// Gives back %true when the node is B-node.

      bool bNode(const Node&) const { return false; }

      /// \brief Gives back the A-node of the undirected edge.

      ///

      /// Gives back the A-node of the undirected edge.

      Node aNode(const UEdge&) const { return INVALID; }

      /// \brief Gives back the B-node of the undirected edge.

      ///

      /// Gives back the B-node of the undirected edge.

      Node bNode(const UEdge&) const { return INVALID; }

      template <typename _Graph>

      struct Constraints {

	typedef typename _Graph::Node Node;

	typedef typename _Graph::ANode ANode;

	typedef typename _Graph::BNode BNode;

	typedef typename _Graph::Edge Edge;

	typedef typename _Graph::UEdge UEdge;

	void constraints() {

          checkConcept<BaseUGraphComponent, _Graph>();

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

	  checkConcept<GraphItem<'b'>, BNode>();

	  {

	    Node n;

	    UEdge ue(INVALID);

            bool b;

	    n = graph.aNode(ue);

	    n = graph.bNode(ue);

            b = graph.aNode(n);

            b = graph.bNode(n);

            ANode an;

            an = n; n = an;

            BNode bn;

            bn = n; n = bn;            

            ignore_unused_variable_warning(b);

	  }      

	}

	const _Graph& graph;

      };

    };

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

    ///  

    /// This class provides beside the core graph features

    /// core id functions for the graph structure.

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

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

    template <typename _Base = BaseGraphComponent>

    class IDableGraphComponent : public _Base {

    public:

      typedef _Base Base;

      typedef typename Base::Node Node;

      typedef typename Base::Edge Edge;

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

      ///

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

      ///

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

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

      ///

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

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

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

      Node nodeFromId(int) const { return INVALID;}

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

      ///

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

      ///

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

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

      ///

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

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

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

      Edge edgeFromId(int) const { return INVALID;}

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

      /// Node id.

      ///

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

      /// id.

      int maxNodeId() const { return -1;}

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

      /// Edge id.

      ///

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

      /// id.

      int maxEdgeId() const { return -1;}

      template <typename _Graph>

      struct Constraints {

	void constraints() {

	  checkConcept<Base, _Graph >();

	  typename _Graph::Node node;

	  int nid = graph.id(node);

	  nid = graph.id(node);

	  node = graph.nodeFromId(nid);

	  typename _Graph::Edge edge;

	  int eid = graph.id(edge);

	  eid = graph.id(edge);

	  edge = graph.edgeFromId(eid);

	  nid = graph.maxNodeId();

	  ignore_unused_variable_warning(nid);

	  eid = graph.maxEdgeId();

	  ignore_unused_variable_warning(eid);

	}

	const _Graph& graph;

      };

    };

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

    ///  

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

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

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

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

    template <typename _Base = BaseUGraphComponent>

    class IDableUGraphComponent : public IDableGraphComponent<_Base> {

    public:

      typedef _Base Base;

      typedef typename Base::UEdge UEdge;

      using IDableGraphComponent<_Base>::id;

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

      ///

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

      ///

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

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

      ///

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

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

      /// result of the function is undetermined.

      UEdge uEdgeFromId(int) const { return INVALID;}

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

      /// UEdge id.

      ///

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

      /// id.

      int maxUEdgeId() const { return -1;}

      template <typename _Graph>

      struct Constraints {

	void constraints() {

	  checkConcept<Base, _Graph >();

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

	  typename _Graph::UEdge uedge;

	  int ueid = graph.id(uedge);

	  ueid = graph.id(uedge);

	  uedge = graph.uEdgeFromId(ueid);

	  ueid = graph.maxUEdgeId();

	  ignore_unused_variable_warning(ueid);

	}

	const _Graph& graph;

      };

    };

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

    ///  

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

    /// features core id functions for the bipartite undirected graph

    /// structure.  The most of the base undirected graphs should be

    /// conform to this concept.

    template <typename _Base = BaseBpUGraphComponent>

    class IDableBpUGraphComponent : public IDableUGraphComponent<_Base> {

    public:

      typedef _Base Base;

      typedef typename Base::Node Node;

      using IDableUGraphComponent<_Base>::id;

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

      ///

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

      ///

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

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

      ///

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

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

      /// result of the function is undetermined.

      Node nodeFromANodeId(int) const { return INVALID;}

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

      /// ANode id.

      ///

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

      /// id.

      int maxANodeId() const { return -1;}

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

      ///

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

      ///

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

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

      ///

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

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

      /// result of the function is undetermined.

      Node nodeFromBNodeId(int) const { return INVALID;}

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

      /// BNode id.

      ///

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

      /// id.

      int maxBNodeId() const { return -1;}

      template <typename _Graph>

      struct Constraints {

	void constraints() {

	  checkConcept<Base, _Graph >();

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

	  typename _Graph::Node node(INVALID);

	  int id;

          id = graph.aNodeId(node);

          id = graph.bNodeId(node);

          node = graph.nodeFromANodeId(id);

          node = graph.nodeFromBNodeId(id);

          id = graph.maxANodeId();

          id = graph.maxBNodeId();

	}

	const _Graph& graph;

      };

    };

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

    ///

    /// Skeleton class for graph NodeIt and EdgeIt.

    ///

    template <typename _Graph, typename _Item>

    class GraphItemIt : public _Item {

    public:

      /// \brief Default constructor.

      ///

      /// @warning The default constructor sets the iterator

      /// to an undefined value.

      GraphItemIt() {}

      /// \brief Copy constructor.

      ///

      /// Copy constructor.

      ///

      GraphItemIt(const GraphItemIt& ) {}

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

      ///

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

      ///

      explicit GraphItemIt(const _Graph&) {}

      /// \brief Invalid constructor \& conversion.

      ///

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

      /// \sa Invalid for more details.

      GraphItemIt(Invalid) {}

      /// \brief Assign operator for items.

      ///

      /// The items are assignable. 

      ///

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

      /// \brief Next item.

      /// 

      /// Assign the iterator to the next item.

      ///

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

      /// \brief Equality operator

      /// 

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

      /// same object or both are invalid.

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

      /// \brief Inequality operator

      ///	

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

      ///

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

      template<typename _GraphItemIt>

      struct Constraints {

	void constraints() {

	  _GraphItemIt it1(g);	

	  _GraphItemIt it2;

	  it2 = ++it1;

	  ++it2 = it1;

	  ++(++it1);

	  _Item bi = it1;

	  bi = it2;

	}

	_Graph& g;

      };

    };

    /// \brief Skeleton class for graph InEdgeIt and OutEdgeIt

    ///

    /// \note Because InEdgeIt and OutEdgeIt may not inherit from the same

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

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

    /// OutEdgeIt with 'o'.

    template <typename _Graph,

	      typename _Item = typename _Graph::Edge,

              typename _Base = typename _Graph::Node, 

	      char _selector = '0'>

    class GraphIncIt : public _Item {

    public:

      /// \brief Default constructor.

      ///

      /// @warning The default constructor sets the iterator

      /// to an undefined value.

      GraphIncIt() {}

      /// \brief Copy constructor.

      ///

      /// Copy constructor.

      ///

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

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

      /// from the node.

      ///

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

      /// from the node.

      ///

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

      /// \brief Invalid constructor \& conversion.

      ///

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

      /// \sa Invalid for more details.

      GraphIncIt(Invalid) {}

      /// \brief Assign operator for iterators.

      ///

      /// The iterators are assignable. 

      ///

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

      /// \brief Next item.

      ///

      /// Assign the iterator to the next item.

      ///

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

      /// \brief Equality operator

      ///

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

      /// same object or both are invalid.

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

      /// \brief Inequality operator

      ///

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

      ///

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

      template <typename _GraphIncIt>

      struct Constraints {

	void constraints() {

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

	  _GraphIncIt it1(graph, node);

	  _GraphIncIt it2;

	  it2 = ++it1;

	  ++it2 = it1;

	  ++(++it1);

	  _Item e = it1;

	  e = it2;

	}

	_Item edge;

	_Base node;

	_Graph graph;

	_GraphIncIt it;

      };

    };

    /// \brief An empty iterable graph class.

    ///

    /// This class provides beside the core graph features

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

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

    template <typename _Base = BaseGraphComponent>

    class IterableGraphComponent : public _Base {

    public:

      typedef _Base Base;

      typedef typename Base::Node Node;

      typedef typename Base::Edge Edge;

      typedef IterableGraphComponent Graph;

      /// \name Base iteration

      /// 

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

      ///

      /// @{  

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

      ///      

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

      ///     

      void first(Node&) const {}

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

      ///

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

      ///     

      void next(Node&) const {}

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

      ///

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

      ///     

      void first(Edge&) const {}

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

      ///

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

      ///     

      void next(Edge&) const {}

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

      /// node.

      ///

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

      ///     

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

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

      /// node.

      ///

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

      ///

      void nextIn(Edge&) const {}

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

      /// given node.

      ///      

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

      ///     

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

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

      /// node.

      ///

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

      ///     

      void nextOut(Edge&) const {}

      /// @}

      /// \name Class based iteration

      /// 

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

      ///

      /// @{

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

      ///

      /// This iterator goes through each node.

      ///

      typedef GraphItemIt<Graph, Node> NodeIt;

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

      ///

      /// This iterator goes through each node.

      ///

      typedef GraphItemIt<Graph, Edge> EdgeIt;

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

      ///

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

      /// of a graph.

      typedef GraphIncIt<Graph, Edge, Node, 'i'> InEdgeIt;

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

      ///

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

      /// of a graph.

      typedef GraphIncIt<Graph, Edge, Node, 'o'> OutEdgeIt;

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

      ///

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

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

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

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

      ///

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

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

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

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

      ///

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

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

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

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

      ///

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

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

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

      /// @}

      template <typename _Graph> 

      struct Constraints {

	void constraints() {

	  checkConcept<Base, _Graph>();

          {

            typename _Graph::Node node(INVALID);      

            typename _Graph::Edge edge(INVALID);

            {

              graph.first(node);

              graph.next(node);

            }

            {

              graph.first(edge);

              graph.next(edge);

            }

            {

              graph.firstIn(edge, node);

              graph.nextIn(edge);

            }

            {

              graph.firstOut(edge, node);

              graph.nextOut(edge);

            }

          }           

          {

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

              typename _Graph::EdgeIt >();

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

              typename _Graph::NodeIt >();

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

              typename _Graph::Node, 'i'>, typename _Graph::InEdgeIt>();

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

              typename _Graph::Node, 'o'>, typename _Graph::OutEdgeIt>();

            typename _Graph::Node n;

            typename _Graph::InEdgeIt ieit(INVALID);

            typename _Graph::OutEdgeIt oeit(INVALID);

            n = graph.baseNode(ieit);

            n = graph.runningNode(ieit);

            n = graph.baseNode(oeit);

            n = graph.runningNode(oeit);

            ignore_unused_variable_warning(n);

          }

        }

	const _Graph& graph;

      };

    };

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

    ///

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

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

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

    template <typename _Base = BaseUGraphComponent>

    class IterableUGraphComponent : public IterableGraphComponent<_Base> {

    public:

      typedef _Base Base;

      typedef typename Base::Node Node;

      typedef typename Base::Edge Edge;

      typedef typename Base::UEdge UEdge;