/* -*- C++ -*-

  *

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

  *

  * Copyright (C) 2003-2006

  * 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 the graph components.

 #ifndef LEMON_CONCEPT_GRAPH_COMPONENTS_H

 #define LEMON_CONCEPT_GRAPH_COMPONENTS_H

 #include <lemon/bits/invalid.h>

 #include <lemon/concept/maps.h>

 #include <lemon/bits/alteration_notifier.h>

 namespace lemon {

   namespace concept {

     /// \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 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 iterable base graph class.

     ///  

     /// This class provides beside the core graph features

     /// core iterable interface for the graph structure.

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

     template <typename _Base = BaseGraphComponent>

     class BaseIterableGraphComponent : public _Base {

     public:

       typedef _Base Base; 

       typedef typename Base::Node Node;

       typedef typename Base::Edge Edge;

       /// \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 {}

       template <typename _Graph>

       struct Constraints {

 	void constraints() {

 	  checkConcept< BaseGraphComponent, _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);

 	  }

 	}

 	const _Graph& graph;

       };

     };

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

     ///  

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

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

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

     /// concept.

     template <typename _Base = BaseUGraphComponent>

     class BaseIterableUGraphComponent 

       : public BaseIterableGraphComponent<_Base> {

     public:

       typedef _Base Base; 

       typedef typename Base::UEdge UEdge;

       typedef typename Base::Node Node;

       using BaseIterableGraphComponent<_Base>::first;

       using BaseIterableGraphComponent<_Base>::next;

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

       /// order.

       ///

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

       ///     

       void first(UEdge&) const {}

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

       /// order.

       ///

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

       ///     

       void next(UEdge&) const {}

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

       /// given node.

       ///

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

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

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

       /// directed edge is the given node.

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

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

       /// given node.

       ///

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

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

       /// use it.

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

       template <typename _Graph>

       struct Constraints {

 	void constraints() {

 	  checkConcept<Base, _Graph >();

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

 	  typename _Graph::Node node(INVALID);

 	  typename _Graph::UEdge uedge(INVALID);

           bool dir;

 	  {

 	    graph.first(uedge);

 	    graph.next(uedge);

 	  }

 	  {

 	    graph.firstInc(uedge, dir, node);

 	    graph.nextInc(uedge, dir);

 	  }

 	}

 	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< BaseGraphComponent, _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 extendable base graph class.

     ///

     /// This class provides beside the core graph features

     /// core graph extend interface for the graph structure.

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

     template <typename _Base = BaseGraphComponent>

     class BaseExtendableGraphComponent : public _Base {

     public:

       typedef typename _Base::Node Node;

       typedef typename _Base::Edge Edge;

       /// \brief Adds a new node to the graph.

       ///

       /// Adds a new node to the graph.

       ///

       Node addNode() {

 	return INVALID;

       }

       /// \brief Adds a new edge connects the given two nodes.

       ///

       /// Adds a new edge connects the the given two nodes.

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

 	return INVALID;

       }

       template <typename _Graph>

       struct Constraints {

 	void constraints() {

 	  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;

       };

     };

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

     ///

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

     /// core undircted graph extend interface for the graph structure.

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

     template <typename _Base = BaseUGraphComponent>

     class BaseExtendableUGraphComponent : public _Base {

     public:

       typedef typename _Base::Node Node;

       typedef typename _Base::UEdge UEdge;

       /// \brief Adds a new node to the graph.

       ///

       /// Adds a new node to the graph.

       ///

       Node addNode() {

 	return INVALID;

       }

       /// \brief Adds a new edge connects the given two nodes.

       ///

       /// Adds a new edge connects the the given two nodes.

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

 	return INVALID;

       }

       template <typename _Graph>

       struct Constraints {

 	void constraints() {

 	  typename _Graph::Node node_a, node_b;

 	  node_a = graph.addNode();

 	  node_b = graph.addNode();

 	  typename _Graph::UEdge uedge;

 	  uedge = graph.addUEdge(node_a, node_b);

 	}

 	_Graph& graph;

       };

     };

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

     ///  

     /// This class provides beside the core graph features

     /// core erase functions for the graph structure.

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

     template <typename _Base = BaseGraphComponent>

     class BaseErasableGraphComponent : public _Base {

     public:

       typedef _Base Base;

       typedef typename Base::Node Node;

       typedef typename Base::Edge Edge;

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

       ///

       /// Erase a node from the graph. This function should not

       /// erase edges connecting to the Node.

       void erase(const Node&) {}    

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

       ///

       /// Erase an edge from the graph.

       ///

       void erase(const Edge&) {}

       template <typename _Graph>

       struct Constraints {

 	void constraints() {

 	  typename _Graph::Node node;

 	  graph.erase(node);

 	  typename _Graph::Edge edge;

 	  graph.erase(edge);

 	}

 	_Graph& graph;

       };

     };

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

     ///  

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

     /// core erase functions for the undirceted graph structure.

     template <typename _Base = BaseUGraphComponent>

     class BaseErasableUGraphComponent : public _Base {

     public:

       typedef _Base Base;

       typedef typename Base::Node Node;

       typedef typename Base::UEdge UEdge;

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

       ///

       /// Erase a node from the graph. This function should not

       /// erase edges connecting to the Node.

       void erase(const Node&) {}    

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

       ///

       /// Erase an edge from the graph.

       ///

       void erase(const UEdge&) {}

       template <typename _Graph>

       struct Constraints {

 	void constraints() {

 	  typename _Graph::Node node;

 	  graph.erase(node);

 	  typename _Graph::Edge edge;

 	  graph.erase(edge);

 	}

 	_Graph& graph;

       };

     };

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

     ///

     /// This class provides beside the core graph features

     /// core clear functions for the graph structure.

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

     template <typename _Base = BaseGraphComponent>

     class BaseClearableGraphComponent : public _Base {

     public:

       /// \brief Erase all the nodes and edges from the graph.

       ///

       /// Erase all the nodes and edges from the graph.

       ///

       void clear() {}    

       template <typename _Graph>

       struct Constraints {

 	void constraints() {

 	  graph.clear();

 	}

 	_Graph graph;

       };

     };

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

     ///

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

     /// core clear functions for the undirected graph structure.

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

     template <typename _Base = BaseUGraphComponent>

     class BaseClearableUGraphComponent : public _Base {

     public:

       /// \brief Erase all the nodes and undirected edges from the graph.

       ///

       /// Erase all the nodes and undirected edges from the graph.

       ///

       void clear() {}    

       template <typename _Graph>

       struct Constraints {

 	void constraints() {

 	  graph.clear();

 	}

 	_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 GraphConcept.

     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;

       /// \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; }

       /// \brief The opposite node on the given edge.

       ///

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

       /// \todo It should not be here.

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

       template <typename _Graph> 

       struct Constraints {

 	void constraints() {

 	  checkConcept<Base, _Graph>();

 	  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 GraphConcept.

     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;

       typedef IterableUGraphComponent Graph;

       using IterableGraphComponent<_Base>::baseNode;

       using IterableGraphComponent<_Base>::runningNode;

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

       ///

       /// This iterator goes through each node.

       typedef GraphItemIt<Graph, UEdge> UEdgeIt;

       /// \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, UEdge, Node, 'u'> IncEdgeIt;

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

       ///

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

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

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

       ///

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

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

       template <typename _Graph> 

       struct Constraints {

 	void constraints() {

 	  checkConcept<Base, _Graph>();

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

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

 	    typename _Graph::UEdgeIt >();

 	  checkConcept<GraphIncIt<_Graph, typename _Graph::UEdge, 

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

           typename _Graph::Node n;

           typename _Graph::IncEdgeIt ueit(INVALID);

           n = graph.baseNode(ueit);

           n = graph.runningNode(ueit);

 	}

 	const _Graph& graph;

       };

     };

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

     ///  

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

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

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

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

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

     /// notified about it.

     template <typename _Base = BaseGraphComponent>

     class AlterableGraphComponent : public _Base {

     public:

       typedef _Base Base;

       typedef typename Base::Node Node;

       typedef typename Base::Edge Edge;

       /// The node observer registry.

       typedef AlterationNotifier<AlterableGraphComponent, Node> 

       NodeNotifier;

       /// The edge observer registry.

       typedef AlterationNotifier<AlterableGraphComponent, Edge> 

       EdgeNotifier;

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

       ///

       /// Gives back the node alteration notifier.

       NodeNotifier& getNotifier(Node) const {

 	return NodeNotifier();

       }

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

       ///

       /// Gives back the edge alteration notifier.

       EdgeNotifier& getNotifier(Edge) const {

 	return EdgeNotifier();

       }

       template <typename _Graph> 

       struct Constraints {

 	void constraints() {

 	  checkConcept<Base, _Graph>();

           typename _Graph::NodeNotifier& nn 

             = graph.getNotifier(typename _Graph::Node());

           typename _Graph::EdgeNotifier& en 

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

           ignore_unused_variable_warning(nn);

           ignore_unused_variable_warning(en);

 	}

 	const _Graph& graph;

       };

     };

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

     ///  

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

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

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

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

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

     /// notified about it.

     template <typename _Base = BaseUGraphComponent>

     class AlterableUGraphComponent : public AlterableGraphComponent<_Base> {

     public:

       typedef _Base Base;

       typedef typename Base::UEdge UEdge;

       /// The edge observer registry.

       typedef AlterationNotifier<AlterableUGraphComponent, UEdge> 

       UEdgeNotifier;

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

       ///

       /// Gives back the edge alteration notifier.

       UEdgeNotifier& getNotifier(UEdge) const {

 	return UEdgeNotifier();

       }

       template <typename _Graph> 

       struct Constraints {

 	void constraints() {

 	  checkConcept<Base, _Graph>();

           checkConcept<AlterableGraphComponent, _Graph>();

           typename _Graph::UEdgeNotifier& uen 

             = graph.getNotifier(typename _Graph::UEdge());

           ignore_unused_variable_warning(uen);

 	}

 	const _Graph& graph;

       };

     };

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

     /// 

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

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

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

     template <typename _Graph, typename _Item, typename _Value>

     class GraphMap : public ReadWriteMap<_Item, _Value> {

     public:

       typedef ReadWriteMap<_Item, _Value> Parent;

       /// The graph type of the map.

       typedef _Graph Graph;

       /// The key type of the map.

       typedef _Item Key;

       /// The value type of the map.

       typedef _Value Value;

       /// \brief Construct a new map.

       ///

       /// Construct a new map for the graph.

       explicit GraphMap(const Graph&) {}

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

       ///

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

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

       /// \brief Copy constructor.

       ///

       /// Copy Constructor.

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

       /// \brief Assign operator.

       ///

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

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

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

       template <typename CMap>

       GraphMap& operator=(const CMap&) { 

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

         return *this;

       }

       template<typename _Map>

       struct Constraints {

 	void constraints() {

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

 	  // Construction with a graph parameter

 	  _Map a(g);

 	  // Constructor with a graph and a default value parameter

 	  _Map a2(g,t);

 	  // Copy constructor.

 	  _Map b(c);

           ReadMap<Key, Value> cmap;

           b = cmap;

 	  ignore_unused_variable_warning(a2);

 	  ignore_unused_variable_warning(b);

 	}

 	const _Map &c;

 	const Graph &g;

 	const typename GraphMap::Value &t;

       };

     };

     /// \brief An empty mappable graph class.

     ///

     /// This class provides beside the core graph features

     /// map interface for the graph structure.

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

     template <typename _Base = BaseGraphComponent>

     class MappableGraphComponent : public _Base  {

     public:

       typedef _Base Base;

       typedef typename Base::Node Node;

       typedef typename Base::Edge Edge;

       typedef MappableGraphComponent Graph;

       /// \brief ReadWrite map of the nodes.

       ///

       /// ReadWrite map of the nodes.

       ///

       template <typename _Value>

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

       private:

 	NodeMap();

       public:

         typedef GraphMap<MappableGraphComponent, Node, _Value> Parent;

 	/// \brief Construct a new map.

 	///

 	/// Construct a new map for the graph.

 	/// \todo call the right parent class constructor

 	explicit NodeMap(const MappableGraphComponent& graph) 

           : Parent(graph) {}

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

 	///

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

 	NodeMap(const MappableGraphComponent& graph, const _Value& value)

           : Parent(graph, value) {}

 	/// \brief Copy constructor.

 	///

 	/// Copy Constructor.

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

 	/// \brief Assign operator.

 	///

 	/// Assign operator.

         template <typename CMap>

         NodeMap& operator=(const CMap&) { 

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

           return *this;

         }

       };

       /// \brief ReadWrite map of the edges.

       ///

       /// ReadWrite map of the edges.

       ///

       template <typename _Value>

       class EdgeMap : public GraphMap<Graph, Edge, _Value> {

       private:

 	EdgeMap();

       public:

         typedef GraphMap<MappableGraphComponent, Edge, _Value> Parent;

 	/// \brief Construct a new map.

 	///

 	/// Construct a new map for the graph.

 	/// \todo call the right parent class constructor

 	explicit EdgeMap(const MappableGraphComponent& graph) 

           : Parent(graph) {}

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

 	///

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

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

           : Parent(graph, value) {}

 	/// \brief Copy constructor.

 	///

 	/// Copy Constructor.

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

 	/// \brief Assign operator.

 	///

 	/// Assign operator.

         template <typename CMap>

         EdgeMap& operator=(const CMap&) { 

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

           return *this;

         }

       };

       template <typename _Graph>

       struct Constraints {

 	struct Dummy {

 	  int value;

 	  Dummy() : value(0) {}

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

 	};

 	void constraints() {

 	  checkConcept<Base, _Graph>();

 	  { // int map test

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

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

 	      IntNodeMap >();

 	  } { // bool map test

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

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

 	      BoolNodeMap >();

 	  } { // Dummy map test

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

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

 	      DummyNodeMap >();

 	  } 

 	  { // 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 >();

 	  } 

 	}

 	_Graph& graph;

       };

     };

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

     ///

     /// This class provides beside the core graph features

     /// map interface for the graph structure.

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

     template <typename _Base = BaseUGraphComponent>

     class MappableUGraphComponent : public MappableGraphComponent<_Base>  {

     public:

       typedef _Base Base;

       typedef typename Base::UEdge UEdge;

       typedef MappableUGraphComponent Graph;

       /// \brief ReadWrite map of the uedges.

       ///

       /// ReadWrite map of the uedges.

       ///

       template <typename _Value>

       class UEdgeMap : public GraphMap<Graph, UEdge, _Value> {  

       public:

         typedef GraphMap<MappableUGraphComponent, UEdge, _Value> Parent;

 	/// \brief Construct a new map.

 	///

 	/// Construct a new map for the graph.

 	/// \todo call the right parent class constructor

 	explicit UEdgeMap(const MappableUGraphComponent& graph) 

           : Parent(graph) {}

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

 	///

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

 	UEdgeMap(const MappableUGraphComponent& graph, const _Value& value)

           : Parent(graph, value) {}

 	/// \brief Copy constructor.

 	///

 	/// Copy Constructor.

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

 	/// \brief Assign operator.

 	///

 	/// Assign operator.

         template <typename CMap>

         UEdgeMap& operator=(const CMap&) { 

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

           return *this;

         }

       };

       template <typename _Graph>

       struct Constraints {

 	struct Dummy {

 	  int value;

 	  Dummy() : value(0) {}

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

 	};

 	void constraints() {

 	  checkConcept<Base, _Graph>();

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

 	  { // int map test

 	    typedef typename _Graph::template UEdgeMap<int> IntUEdgeMap;

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

 	      IntUEdgeMap >();

 	  } { // bool map test

 	    typedef typename _Graph::template UEdgeMap<bool> BoolUEdgeMap;

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

 	      BoolUEdgeMap >();

 	  } { // Dummy map test

 	    typedef typename _Graph::template UEdgeMap<Dummy> DummyUEdgeMap;

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

 	      DummyUEdgeMap >();

 	  } 

 	}

 	_Graph& graph;

       };

     };

     /// \brief An empty extendable graph class.

     ///

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

     /// extendable interface for the graph structure.  The main

     /// difference between the base and this interface is that the

     /// graph alterations should handled already on this level.

     template <typename _Base = BaseGraphComponent>

     class ExtendableGraphComponent : public _Base {

     public:

       typedef typename _Base::Node Node;

       typedef typename _Base::Edge Edge;

       /// \brief Adds a new node to the graph.

       ///

       /// Adds a new node to the graph.

       ///

       Node addNode() {

 	return INVALID;

       }

       /// \brief Adds a new edge connects the given two nodes.

       ///

       /// Adds a new edge connects the the given two nodes.

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

 	return INVALID;

       }

       template <typename _Graph>

       struct Constraints {

 	void constraints() {

 	  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;

       };

     };

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

     ///

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

     /// core undircted graph extend interface for the graph structure.

     /// The main difference between the base and this interface is

     /// that the graph alterations should handled already on this

     /// level.

     template <typename _Base = BaseUGraphComponent>

     class ExtendableUGraphComponent : public _Base {

     public:

       typedef typename _Base::Node Node;

       typedef typename _Base::UEdge UEdge;

       /// \brief Adds a new node to the graph.

       ///

       /// Adds a new node to the graph.

       ///

       Node addNode() {

 	return INVALID;

       }

       /// \brief Adds a new edge connects the given two nodes.

       ///

       /// Adds a new edge connects the the given two nodes.

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

 	return INVALID;

       }

       template <typename _Graph>

       struct Constraints {

 	void constraints() {

 	  typename _Graph::Node node_a, node_b;

 	  node_a = graph.addNode();

 	  node_b = graph.addNode();

 	  typename _Graph::UEdge uedge;

 	  uedge = graph.addUEdge(node_a, node_b);

 	}

 	_Graph& graph;

       };

     };

     /// \brief An empty erasable graph class.

     ///  

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

     /// functions for the graph structure. The main difference between

     /// the base and this interface is that the graph alterations

     /// should handled already on this level.

     template <typename _Base = BaseGraphComponent>

     class ErasableGraphComponent : public _Base {

     public:

       typedef _Base Base;

       typedef typename Base::Node Node;

       typedef typename Base::Edge Edge;

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

       ///

       /// Erase a node from the graph. This function should 

       /// erase all edges connecting to the node.

       void erase(const Node&) {}    

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

       ///

       /// Erase an edge from the graph.

       ///

       void erase(const Edge&) {}

       template <typename _Graph>

       struct Constraints {

 	void constraints() {

 	  typename _Graph::Node node;

 	  graph.erase(node);

 	  typename _Graph::Edge edge;

 	  graph.erase(edge);

 	}

 	_Graph& graph;

       };

     };

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

     ///  

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

     /// core erase functions for the undirceted graph structure. The

     /// main difference between the base and this interface is that

     /// the graph alterations should handled already on this level.

     template <typename _Base = BaseUGraphComponent>

     class ErasableUGraphComponent : public _Base {

     public:

       typedef _Base Base;

       typedef typename Base::Node Node;

       typedef typename Base::UEdge UEdge;

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

       ///

       /// Erase a node from the graph. This function should erase

       /// edges connecting to the node.

       void erase(const Node&) {}    

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

       ///

       /// Erase an edge from the graph.

       ///

       void erase(const UEdge&) {}

       template <typename _Graph>

       struct Constraints {

 	void constraints() {

 	  typename _Graph::Node node;

 	  graph.erase(node);

 	  typename _Graph::Edge edge;

 	  graph.erase(edge);

 	}

 	_Graph& graph;

       };

     };

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

     ///

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

     /// functions for the graph structure. The main difference between

     /// the base and this interface is that the graph alterations

     /// should handled already on this level.

     template <typename _Base = BaseGraphComponent>

     class ClearableGraphComponent : public _Base {

     public:

       /// \brief Erase all nodes and edges from the graph.

       ///

       /// Erase all nodes and edges from the graph.

       ///

       void clear() {}    

       template <typename _Graph>

       struct Constraints {

 	void constraints() {

 	  graph.clear();

 	}

 	_Graph graph;

       };

     };

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

     ///

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

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

     /// main difference between the base and this interface is that

     /// the graph alterations should handled already on this level.

     template <typename _Base = BaseUGraphComponent>

     class ClearableUGraphComponent : public _Base {

     public:

       /// \brief Erase all nodes and undirected edges from the graph.

       ///

       /// Erase all nodes and undirected edges from the graph.

       ///

       void clear() {}    

       template <typename _Graph>

       struct Constraints {

 	void constraints() {

 	  graph.clear();

 	}

 	_Graph graph;

       };

     };

   }

 }

 #endif