diff --git a/lemon/concepts/graph_components.h b/lemon/concepts/graph_components.h
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+/* -*- C++ -*-
+ *
+ * This file is a part of LEMON, a generic C++ optimization library
+ *
+ * Copyright (C) 2003-2007
+ * Egervary Jeno Kombinatorikus Optimalizalasi Kutatocsoport
+ * (Egervary Research Group on Combinatorial Optimization, EGRES).
+ *
+ * Permission to use, modify and distribute this software is granted
+ * provided that this copyright notice appears in all copies. For
+ * precise terms see the accompanying LICENSE file.
+ *
+ * This software is provided "AS IS" with no warranty of any kind,
+ * express or implied, and with no claim as to its suitability for any
+ * purpose.
+ *
+ */
+
+///\ingroup graph_concepts
+///\file
+///\brief The concept of graph components.
+
+
+#ifndef LEMON_CONCEPT_GRAPH_COMPONENTS_H
+#define LEMON_CONCEPT_GRAPH_COMPONENTS_H
+
+#include <lemon/bits/invalid.h>
+#include <lemon/concepts/maps.h>
+
+#include <lemon/bits/alteration_notifier.h>
+
+namespace lemon {
+  namespace concepts {
+
+    /// \brief Skeleton class for graph Node and Arc types
+    ///
+    /// This class describes the interface of Node and Arc (and Edge
+    /// in undirected graphs) subtypes of graph types.
+    ///
+    /// \note This class is a template class so that we can use it to
+    /// create graph skeleton classes. The reason for this is than Node
+    /// and Arc types should \em not derive from the same base class.
+    /// For Node you should instantiate it with character 'n' and for Arc
+    /// with 'a'.
+
+#ifndef DOXYGEN
+    template <char _selector = '0'>
+#endif
+    class GraphItem {
+    public:
+      /// \brief Default constructor.
+      ///      
+      /// \warning The default constructor is not required to set
+      /// the item to some well-defined value. So you should consider it
+      /// as uninitialized.
+      GraphItem() {}
+      /// \brief Copy constructor.
+      ///
+      /// Copy constructor.
+      ///
+      GraphItem(const GraphItem &) {}
+      /// \brief Invalid constructor \& conversion.
+      ///
+      /// This constructor initializes the item to be invalid.
+      /// \sa Invalid for more details.
+      GraphItem(Invalid) {}
+      /// \brief Assign operator for nodes.
+      ///
+      /// The nodes are assignable. 
+      ///
+      GraphItem& operator=(GraphItem const&) { return *this; }
+      /// \brief Equality operator.
+      ///
+      /// Two iterators are equal if and only if they represents the
+      /// same node in the graph or both are invalid.
+      bool operator==(GraphItem) const { return false; }
+      /// \brief Inequality operator.
+      ///
+      /// \sa operator==(const Node& n)
+      ///
+      bool operator!=(GraphItem) const { return false; }
+
+      /// \brief Artificial ordering operator.
+      ///
+      /// To allow the use of graph descriptors as key type in std::map or
+      /// similar associative container we require this.
+      ///
+      /// \note This operator only have to define some strict ordering of
+      /// the items; this order has nothing to do with the iteration
+      /// ordering of the items.
+      bool operator<(GraphItem) const { return false; }
+
+      template<typename _GraphItem>
+      struct Constraints {
+	void constraints() {
+	  _GraphItem i1;
+	  _GraphItem i2 = i1;
+	  _GraphItem i3 = INVALID;
+	  
+	  i1 = i2 = i3;
+
+	  bool b;
+	  //	  b = (ia == ib) && (ia != ib) && (ia < ib);
+	  b = (ia == ib) && (ia != ib);
+	  b = (ia == INVALID) && (ib != INVALID);
+          b = (ia < ib);
+	}
+
+	const _GraphItem &ia;
+	const _GraphItem &ib;
+      };
+    };
+
+    /// \brief An empty base directed graph class.
+    ///  
+    /// This class provides the minimal set of features needed for a
+    /// directed graph structure. All digraph concepts have to be
+    /// conform to this base directed graph. It just provides types
+    /// for nodes and arcs and functions to get the source and the
+    /// target of the arcs.
+    class BaseDigraphComponent {
+    public:
+
+      typedef BaseDigraphComponent Digraph;
+      
+      /// \brief Node class of the digraph.
+      ///
+      /// This class represents the Nodes of the digraph. 
+      ///
+      typedef GraphItem<'n'> Node;
+
+      /// \brief Arc class of the digraph.
+      ///
+      /// This class represents the Arcs of the digraph. 
+      ///
+      typedef GraphItem<'e'> Arc;
+
+      /// \brief Gives back the target node of an arc.
+      ///
+      /// Gives back the target node of an arc.
+      ///
+      Node target(const Arc&) const { return INVALID;}
+
+      /// \brief Gives back the source node of an arc.
+      ///
+      /// Gives back the source node of an arc.
+      ///
+      Node source(const Arc&) const { return INVALID;}
+
+      /// \brief Gives back the opposite node on the given arc.
+      ///
+      /// Gives back the opposite node on the given arc.
+      Node oppositeNode(const Node&, const Arc&) const {
+        return INVALID;
+      }
+
+      template <typename _Digraph>
+      struct Constraints {
+	typedef typename _Digraph::Node Node;
+	typedef typename _Digraph::Arc Arc;
+      
+	void constraints() {
+	  checkConcept<GraphItem<'n'>, Node>();
+	  checkConcept<GraphItem<'a'>, Arc>();
+	  {
+	    Node n;
+	    Arc e(INVALID);
+	    n = digraph.source(e);
+	    n = digraph.target(e);
+            n = digraph.oppositeNode(n, e);
+	  }      
+	}
+      
+	const _Digraph& digraph;
+      };
+    };
+
+    /// \brief An empty base undirected graph class.
+    ///  
+    /// This class provides the minimal set of features needed for an
+    /// undirected graph structure. All undirected graph concepts have
+    /// to be conform to this base graph. It just provides types for
+    /// nodes, arcs and edges and functions to get the
+    /// source and the target of the arcs and edges,
+    /// conversion from arcs to edges and function to get
+    /// both direction of the edges.
+    class BaseGraphComponent : public BaseDigraphComponent {
+    public:
+      typedef BaseDigraphComponent::Node Node;
+      typedef BaseDigraphComponent::Arc Arc;
+      /// \brief Undirected arc class of the graph.
+      ///
+      /// This class represents the edges of the graph.
+      /// The undirected graphs can be used as a directed graph which
+      /// for each arc contains the opposite arc too so the graph is
+      /// bidirected. The edge represents two opposite
+      /// directed arcs.
+      class Edge : public GraphItem<'u'> {
+      public:
+        typedef GraphItem<'u'> Parent;
+        /// \brief Default constructor.
+        ///      
+        /// \warning The default constructor is not required to set
+        /// the item to some well-defined value. So you should consider it
+        /// as uninitialized.
+        Edge() {}
+        /// \brief Copy constructor.
+        ///
+        /// Copy constructor.
+        ///
+        Edge(const Edge &) : Parent() {}
+        /// \brief Invalid constructor \& conversion.
+        ///
+        /// This constructor initializes the item to be invalid.
+        /// \sa Invalid for more details.
+        Edge(Invalid) {}
+        /// \brief Converter from arc to edge.
+        ///
+        /// Besides the core graph item functionality each arc should
+        /// be convertible to the represented edge. 
+        Edge(const Arc&) {}
+        /// \brief Assign arc to edge.
+        ///
+        /// Besides the core graph item functionality each arc should
+        /// be convertible to the represented edge. 
+        Edge& operator=(const Arc&) { return *this; }
+      };
+
+      /// \brief Returns the direction of the arc.
+      ///
+      /// Returns the direction of the arc. Each arc represents an
+      /// edge with a direction. It gives back the
+      /// direction.
+      bool direction(const Arc&) const { return true; }
+
+      /// \brief Returns the directed arc.
+      ///
+      /// Returns the directed arc from its direction and the
+      /// represented edge.
+      Arc direct(const Edge&, bool) const { return INVALID;} 
+
+      /// \brief Returns the directed arc.
+      ///
+      /// Returns the directed arc from its source and the
+      /// represented edge.
+      Arc direct(const Edge&, const Node&) const { return INVALID;} 
+
+      /// \brief Returns the opposite arc.
+      ///
+      /// Returns the opposite arc. It is the arc representing the
+      /// same edge and has opposite direction.
+      Arc oppositeArc(const Arc&) const { return INVALID;}
+
+      /// \brief Gives back one ending of an edge.
+      ///
+      /// Gives back one ending of an edge.
+      Node u(const Edge&) const { return INVALID;}
+
+      /// \brief Gives back the other ending of an edge.
+      ///
+      /// Gives back the other ending of an edge.
+      Node v(const Edge&) const { return INVALID;}
+      
+      template <typename _Graph>
+      struct Constraints {
+	typedef typename _Graph::Node Node;
+	typedef typename _Graph::Arc Arc;
+	typedef typename _Graph::Edge Edge;
+      
+	void constraints() {
+          checkConcept<BaseDigraphComponent, _Graph>();
+	  checkConcept<GraphItem<'u'>, Edge>();
+	  {
+	    Node n;
+	    Edge ue(INVALID);
+            Arc e;
+	    n = graph.u(ue);
+	    n = graph.v(ue);
+            e = graph.direct(ue, true);
+            e = graph.direct(ue, n);
+            e = graph.oppositeArc(e);
+            ue = e;
+            bool d = graph.direction(e);
+            ignore_unused_variable_warning(d);
+	  }      
+	}
+      
+	const _Graph& graph;
+      };
+
+    };
+
+    /// \brief An empty idable base digraph class.
+    ///  
+    /// This class provides beside the core digraph features
+    /// core id functions for the digraph structure.
+    /// The most of the base digraphs should be conform to this concept.
+    /// The id's are unique and immutable.
+    template <typename _Base = BaseDigraphComponent>
+    class IDableDigraphComponent : public _Base {
+    public:
+
+      typedef _Base Base;
+      typedef typename Base::Node Node;
+      typedef typename Base::Arc Arc;
+
+      /// \brief Gives back an unique integer id for the Node. 
+      ///
+      /// Gives back an unique integer id for the Node. 
+      ///
+      int id(const Node&) const { return -1;}
+
+      /// \brief Gives back the node by the unique id.
+      ///
+      /// Gives back the node by the unique id.
+      /// If the digraph does not contain node with the given id
+      /// then the result of the function is undetermined. 
+      Node nodeFromId(int) const { return INVALID;}
+
+      /// \brief Gives back an unique integer id for the Arc. 
+      ///
+      /// Gives back an unique integer id for the Arc. 
+      ///
+      int id(const Arc&) const { return -1;}
+
+      /// \brief Gives back the arc by the unique id.
+      ///
+      /// Gives back the arc by the unique id.
+      /// If the digraph does not contain arc with the given id
+      /// then the result of the function is undetermined. 
+      Arc arcFromId(int) const { return INVALID;}
+
+      /// \brief Gives back an integer greater or equal to the maximum
+      /// Node id.
+      ///
+      /// Gives back an integer greater or equal to the maximum Node
+      /// id.
+      int maxNodeId() const { return -1;}
+
+      /// \brief Gives back an integer greater or equal to the maximum
+      /// Arc id.
+      ///
+      /// Gives back an integer greater or equal to the maximum Arc
+      /// id.
+      int maxArcId() const { return -1;}
+
+      template <typename _Digraph>
+      struct Constraints {
+
+	void constraints() {
+	  checkConcept<Base, _Digraph >();
+	  typename _Digraph::Node node;
+	  int nid = digraph.id(node);
+	  nid = digraph.id(node);
+	  node = digraph.nodeFromId(nid);
+	  typename _Digraph::Arc arc;
+	  int eid = digraph.id(arc);
+	  eid = digraph.id(arc);
+	  arc = digraph.arcFromId(eid);
+
+	  nid = digraph.maxNodeId();
+	  ignore_unused_variable_warning(nid);
+	  eid = digraph.maxArcId();
+	  ignore_unused_variable_warning(eid);
+	}
+
+	const _Digraph& digraph;
+      };
+    };
+
+    /// \brief An empty idable base undirected graph class.
+    ///  
+    /// This class provides beside the core undirected graph features
+    /// core id functions for the undirected graph structure.  The
+    /// most of the base undirected graphs should be conform to this
+    /// concept.  The id's are unique and immutable.
+    template <typename _Base = BaseGraphComponent>
+    class IDableGraphComponent : public IDableDigraphComponent<_Base> {
+    public:
+
+      typedef _Base Base;
+      typedef typename Base::Edge Edge;
+
+      using IDableDigraphComponent<_Base>::id;
+
+      /// \brief Gives back an unique integer id for the Edge. 
+      ///
+      /// Gives back an unique integer id for the Edge. 
+      ///
+      int id(const Edge&) const { return -1;}
+
+      /// \brief Gives back the edge by the unique id.
+      ///
+      /// Gives back the edge by the unique id.  If the
+      /// graph does not contain arc with the given id then the
+      /// result of the function is undetermined.
+      Edge edgeFromId(int) const { return INVALID;}
+
+      /// \brief Gives back an integer greater or equal to the maximum
+      /// Edge id.
+      ///
+      /// Gives back an integer greater or equal to the maximum Edge
+      /// id.
+      int maxEdgeId() const { return -1;}
+
+      template <typename _Graph>
+      struct Constraints {
+
+	void constraints() {
+	  checkConcept<Base, _Graph >();
+	  checkConcept<IDableDigraphComponent<Base>, _Graph >();
+	  typename _Graph::Edge edge;
+	  int ueid = graph.id(edge);
+	  ueid = graph.id(edge);
+	  edge = graph.edgeFromId(ueid);
+	  ueid = graph.maxEdgeId();
+	  ignore_unused_variable_warning(ueid);
+	}
+
+	const _Graph& graph;
+      };
+    };
+
+    /// \brief Skeleton class for graph NodeIt and ArcIt
+    ///
+    /// Skeleton class for graph NodeIt and ArcIt.
+    ///
+    template <typename _Graph, typename _Item>
+    class GraphItemIt : public _Item {
+    public:
+      /// \brief Default constructor.
+      ///
+      /// @warning The default constructor sets the iterator
+      /// to an undefined value.
+      GraphItemIt() {}
+      /// \brief Copy constructor.
+      ///
+      /// Copy constructor.
+      ///
+      GraphItemIt(const GraphItemIt& ) {}
+      /// \brief Sets the iterator to the first item.
+      ///
+      /// Sets the iterator to the first item of \c the graph.
+      ///
+      explicit GraphItemIt(const _Graph&) {}
+      /// \brief Invalid constructor \& conversion.
+      ///
+      /// This constructor initializes the item to be invalid.
+      /// \sa Invalid for more details.
+      GraphItemIt(Invalid) {}
+      /// \brief Assign operator for items.
+      ///
+      /// The items are assignable. 
+      ///
+      GraphItemIt& operator=(const GraphItemIt&) { return *this; }      
+      /// \brief Next item.
+      /// 
+      /// Assign the iterator to the next item.
+      ///
+      GraphItemIt& operator++() { return *this; }
+      /// \brief Equality operator
+      /// 
+      /// Two iterators are equal if and only if they point to the
+      /// same object or both are invalid.
+      bool operator==(const GraphItemIt&) const { return true;}
+      /// \brief Inequality operator
+      ///	
+      /// \sa operator==(Node n)
+      ///
+      bool operator!=(const GraphItemIt&) const { return true;}
+      
+      template<typename _GraphItemIt>
+      struct Constraints {
+	void constraints() {
+	  _GraphItemIt it1(g);	
+	  _GraphItemIt it2;
+
+	  it2 = ++it1;
+	  ++it2 = it1;
+	  ++(++it1);
+
+	  _Item bi = it1;
+	  bi = it2;
+	}
+	_Graph& g;
+      };
+    };
+
+    /// \brief Skeleton class for graph InArcIt and OutArcIt
+    ///
+    /// \note Because InArcIt and OutArcIt may not inherit from the same
+    /// base class, the _selector is a additional template parameter. For 
+    /// InArcIt you should instantiate it with character 'i' and for 
+    /// OutArcIt with 'o'.
+    template <typename _Graph,
+	      typename _Item = typename _Graph::Arc,
+              typename _Base = typename _Graph::Node, 
+	      char _selector = '0'>
+    class GraphIncIt : public _Item {
+    public:
+      /// \brief Default constructor.
+      ///
+      /// @warning The default constructor sets the iterator
+      /// to an undefined value.
+      GraphIncIt() {}
+      /// \brief Copy constructor.
+      ///
+      /// Copy constructor.
+      ///
+      GraphIncIt(GraphIncIt const& gi) : _Item(gi) {}
+      /// \brief Sets the iterator to the first arc incoming into or outgoing 
+      /// from the node.
+      ///
+      /// Sets the iterator to the first arc incoming into or outgoing 
+      /// from the node.
+      ///
+      explicit GraphIncIt(const _Graph&, const _Base&) {}
+      /// \brief Invalid constructor \& conversion.
+      ///
+      /// This constructor initializes the item to be invalid.
+      /// \sa Invalid for more details.
+      GraphIncIt(Invalid) {}
+      /// \brief Assign operator for iterators.
+      ///
+      /// The iterators are assignable. 
+      ///
+      GraphIncIt& operator=(GraphIncIt const&) { return *this; }      
+      /// \brief Next item.
+      ///
+      /// Assign the iterator to the next item.
+      ///
+      GraphIncIt& operator++() { return *this; }
+
+      /// \brief Equality operator
+      ///
+      /// Two iterators are equal if and only if they point to the
+      /// same object or both are invalid.
+      bool operator==(const GraphIncIt&) const { return true;}
+
+      /// \brief Inequality operator
+      ///
+      /// \sa operator==(Node n)
+      ///
+      bool operator!=(const GraphIncIt&) const { return true;}
+
+      template <typename _GraphIncIt>
+      struct Constraints {
+	void constraints() {
+	  checkConcept<GraphItem<_selector>, _GraphIncIt>();
+	  _GraphIncIt it1(graph, node);
+	  _GraphIncIt it2;
+
+	  it2 = ++it1;
+	  ++it2 = it1;
+	  ++(++it1);
+	  _Item e = it1;
+	  e = it2;
+
+	}
+
+	_Item arc;
+	_Base node;
+	_Graph graph;
+	_GraphIncIt it;
+      };
+    };
+
+
+    /// \brief An empty iterable digraph class.
+    ///
+    /// This class provides beside the core digraph features
+    /// iterator based iterable interface for the digraph structure.
+    /// This concept is part of the Digraph concept.
+    template <typename _Base = BaseDigraphComponent>
+    class IterableDigraphComponent : public _Base {
+
+    public:
+    
+      typedef _Base Base;
+      typedef typename Base::Node Node;
+      typedef typename Base::Arc Arc;
+
+      typedef IterableDigraphComponent Digraph;
+
+      /// \name Base iteration
+      /// 
+      /// This interface provides functions for iteration on digraph items
+      ///
+      /// @{  
+
+      /// \brief Gives back the first node in the iterating order.
+      ///      
+      /// Gives back the first node in the iterating order.
+      ///     
+      void first(Node&) const {}
+
+      /// \brief Gives back the next node in the iterating order.
+      ///
+      /// Gives back the next node in the iterating order.
+      ///     
+      void next(Node&) const {}
+
+      /// \brief Gives back the first arc in the iterating order.
+      ///
+      /// Gives back the first arc in the iterating order.
+      ///     
+      void first(Arc&) const {}
+
+      /// \brief Gives back the next arc in the iterating order.
+      ///
+      /// Gives back the next arc in the iterating order.
+      ///     
+      void next(Arc&) const {}
+
+
+      /// \brief Gives back the first of the arcs point to the given
+      /// node.
+      ///
+      /// Gives back the first of the arcs point to the given node.
+      ///     
+      void firstIn(Arc&, const Node&) const {}
+
+      /// \brief Gives back the next of the arcs points to the given
+      /// node.
+      ///
+      /// Gives back the next of the arcs points to the given node.
+      ///
+      void nextIn(Arc&) const {}
+
+      /// \brief Gives back the first of the arcs start from the
+      /// given node.
+      ///      
+      /// Gives back the first of the arcs start from the given node.
+      ///     
+      void firstOut(Arc&, const Node&) const {}
+
+      /// \brief Gives back the next of the arcs start from the given
+      /// node.
+      ///
+      /// Gives back the next of the arcs start from the given node.
+      ///     
+      void nextOut(Arc&) const {}
+
+      /// @}
+
+      /// \name Class based iteration
+      /// 
+      /// This interface provides functions for iteration on digraph items
+      ///
+      /// @{
+
+      /// \brief This iterator goes through each node.
+      ///
+      /// This iterator goes through each node.
+      ///
+      typedef GraphItemIt<Digraph, Node> NodeIt;
+
+      /// \brief This iterator goes through each node.
+      ///
+      /// This iterator goes through each node.
+      ///
+      typedef GraphItemIt<Digraph, Arc> ArcIt;
+
+      /// \brief This iterator goes trough the incoming arcs of a node.
+      ///
+      /// This iterator goes trough the \e inccoming arcs of a certain node
+      /// of a digraph.
+      typedef GraphIncIt<Digraph, Arc, Node, 'i'> InArcIt;
+
+      /// \brief This iterator goes trough the outgoing arcs of a node.
+      ///
+      /// This iterator goes trough the \e outgoing arcs of a certain node
+      /// of a digraph.
+      typedef GraphIncIt<Digraph, Arc, Node, 'o'> OutArcIt;
+
+      /// \brief The base node of the iterator.
+      ///
+      /// Gives back the base node of the iterator.
+      /// It is always the target of the pointed arc.
+      Node baseNode(const InArcIt&) const { return INVALID; }
+
+      /// \brief The running node of the iterator.
+      ///
+      /// Gives back the running node of the iterator.
+      /// It is always the source of the pointed arc.
+      Node runningNode(const InArcIt&) const { return INVALID; }
+
+      /// \brief The base node of the iterator.
+      ///
+      /// Gives back the base node of the iterator.
+      /// It is always the source of the pointed arc.
+      Node baseNode(const OutArcIt&) const { return INVALID; }
+
+      /// \brief The running node of the iterator.
+      ///
+      /// Gives back the running node of the iterator.
+      /// It is always the target of the pointed arc.
+      Node runningNode(const OutArcIt&) const { return INVALID; }
+
+      /// @}
+
+      template <typename _Digraph> 
+      struct Constraints {
+	void constraints() {
+	  checkConcept<Base, _Digraph>();
+
+          {
+            typename _Digraph::Node node(INVALID);      
+            typename _Digraph::Arc arc(INVALID);
+            {
+              digraph.first(node);
+              digraph.next(node);
+            }
+            {
+              digraph.first(arc);
+              digraph.next(arc);
+            }
+            {
+              digraph.firstIn(arc, node);
+              digraph.nextIn(arc);
+            }
+            {
+              digraph.firstOut(arc, node);
+              digraph.nextOut(arc);
+            }
+          }           
+
+          {
+            checkConcept<GraphItemIt<_Digraph, typename _Digraph::Arc>,
+              typename _Digraph::ArcIt >();
+            checkConcept<GraphItemIt<_Digraph, typename _Digraph::Node>,
+              typename _Digraph::NodeIt >();
+            checkConcept<GraphIncIt<_Digraph, typename _Digraph::Arc, 
+              typename _Digraph::Node, 'i'>, typename _Digraph::InArcIt>();
+            checkConcept<GraphIncIt<_Digraph, typename _Digraph::Arc, 
+              typename _Digraph::Node, 'o'>, typename _Digraph::OutArcIt>();
+
+            typename _Digraph::Node n;
+            typename _Digraph::InArcIt ieit(INVALID);
+            typename _Digraph::OutArcIt oeit(INVALID);
+            n = digraph.baseNode(ieit);
+            n = digraph.runningNode(ieit);
+            n = digraph.baseNode(oeit);
+            n = digraph.runningNode(oeit);
+            ignore_unused_variable_warning(n);
+          }
+        }
+	
+	const _Digraph& digraph;
+	
+      };
+    };
+
+    /// \brief An empty iterable undirected graph class.
+    ///
+    /// This class provides beside the core graph features iterator
+    /// based iterable interface for the undirected graph structure.
+    /// This concept is part of the Graph concept.
+    template <typename _Base = BaseGraphComponent>
+    class IterableGraphComponent : public IterableDigraphComponent<_Base> {
+    public:
+
+      typedef _Base Base;
+      typedef typename Base::Node Node;
+      typedef typename Base::Arc Arc;
+      typedef typename Base::Edge Edge;
+
+    
+      typedef IterableGraphComponent Graph;
+
+      /// \name Base iteration
+      /// 
+      /// This interface provides functions for iteration on graph items
+      /// @{  
+
+      using IterableDigraphComponent<_Base>::first;
+      using IterableDigraphComponent<_Base>::next;
+
+      /// \brief Gives back the first edge in the iterating
+      /// order.
+      ///
+      /// Gives back the first edge in the iterating order.
+      ///     
+      void first(Edge&) const {}
+
+      /// \brief Gives back the next edge in the iterating
+      /// order.
+      ///
+      /// Gives back the next edge in the iterating order.
+      ///     
+      void next(Edge&) const {}
+
+
+      /// \brief Gives back the first of the edges from the
+      /// given node.
+      ///
+      /// Gives back the first of the edges from the given
+      /// node. The bool parameter gives back that direction which
+      /// gives a good direction of the edge so the source of the
+      /// directed arc is the given node.
+      void firstInc(Edge&, bool&, const Node&) const {}
+
+      /// \brief Gives back the next of the edges from the
+      /// given node.
+      ///
+      /// Gives back the next of the edges from the given
+      /// node. The bool parameter should be used as the \c firstInc()
+      /// use it.
+      void nextInc(Edge&, bool&) const {}
+
+      using IterableDigraphComponent<_Base>::baseNode;
+      using IterableDigraphComponent<_Base>::runningNode;
+
+      /// @}
+
+      /// \name Class based iteration
+      /// 
+      /// This interface provides functions for iteration on graph items
+      ///
+      /// @{
+
+      /// \brief This iterator goes through each node.
+      ///
+      /// This iterator goes through each node.
+      typedef GraphItemIt<Graph, Edge> EdgeIt;
+      /// \brief This iterator goes trough the incident arcs of a
+      /// node.
+      ///
+      /// This iterator goes trough the incident arcs of a certain
+      /// node of a graph.
+      typedef GraphIncIt<Graph, Edge, Node, 'u'> IncArcIt;
+      /// \brief The base node of the iterator.
+      ///
+      /// Gives back the base node of the iterator.
+      Node baseNode(const IncArcIt&) const { return INVALID; }
+
+      /// \brief The running node of the iterator.
+      ///
+      /// Gives back the running node of the iterator.
+      Node runningNode(const IncArcIt&) const { return INVALID; }
+
+      /// @}
+
+      template <typename _Graph> 
+      struct Constraints {
+	void constraints() {
+	  checkConcept<IterableDigraphComponent<Base>, _Graph>();
+
+          {
+            typename _Graph::Node node(INVALID);
+            typename _Graph::Edge edge(INVALID);
+            bool dir;
+            {
+              graph.first(edge);
+              graph.next(edge);
+            }
+            {
+              graph.firstInc(edge, dir, node);
+              graph.nextInc(edge, dir);
+            }
+            
+          }	
+  
+          {
+            checkConcept<GraphItemIt<_Graph, typename _Graph::Edge>,
+              typename _Graph::EdgeIt >();
+            checkConcept<GraphIncIt<_Graph, typename _Graph::Edge, 
+              typename _Graph::Node, 'u'>, typename _Graph::IncArcIt>();
+            
+            typename _Graph::Node n;
+            typename _Graph::IncArcIt ueit(INVALID);
+            n = graph.baseNode(ueit);
+            n = graph.runningNode(ueit);
+          }
+        }
+	
+	const _Graph& graph;
+	
+      };
+    };
+
+    /// \brief An empty alteration notifier digraph class.
+    ///  
+    /// This class provides beside the core digraph features alteration
+    /// notifier interface for the digraph structure.  This implements
+    /// an observer-notifier pattern for each digraph item. More
+    /// obsevers can be registered into the notifier and whenever an
+    /// alteration occured in the digraph all the observers will
+    /// notified about it.
+    template <typename _Base = BaseDigraphComponent>
+    class AlterableDigraphComponent : public _Base {
+    public:
+
+      typedef _Base Base;
+      typedef typename Base::Node Node;
+      typedef typename Base::Arc Arc;
+
+
+      /// The node observer registry.
+      typedef AlterationNotifier<AlterableDigraphComponent, Node> 
+      NodeNotifier;
+      /// The arc observer registry.
+      typedef AlterationNotifier<AlterableDigraphComponent, Arc> 
+      ArcNotifier;
+      
+      /// \brief Gives back the node alteration notifier.
+      ///
+      /// Gives back the node alteration notifier.
+      NodeNotifier& notifier(Node) const {
+	return NodeNotifier();
+      }
+      
+      /// \brief Gives back the arc alteration notifier.
+      ///
+      /// Gives back the arc alteration notifier.
+      ArcNotifier& notifier(Arc) const {
+	return ArcNotifier();
+      }
+
+      template <typename _Digraph> 
+      struct Constraints {
+	void constraints() {
+	  checkConcept<Base, _Digraph>();
+          typename _Digraph::NodeNotifier& nn 
+            = digraph.notifier(typename _Digraph::Node());
+
+          typename _Digraph::ArcNotifier& en 
+            = digraph.notifier(typename _Digraph::Arc());
+          
+          ignore_unused_variable_warning(nn);
+          ignore_unused_variable_warning(en);
+	}
+	
+	const _Digraph& digraph;
+	
+      };
+      
+    };
+
+    /// \brief An empty alteration notifier undirected graph class.
+    ///  
+    /// This class provides beside the core graph features alteration
+    /// notifier interface for the graph structure.  This implements
+    /// an observer-notifier pattern for each graph item. More
+    /// obsevers can be registered into the notifier and whenever an
+    /// alteration occured in the graph all the observers will
+    /// notified about it.
+    template <typename _Base = BaseGraphComponent>
+    class AlterableGraphComponent : public AlterableDigraphComponent<_Base> {
+    public:
+
+      typedef _Base Base;
+      typedef typename Base::Edge Edge;
+
+
+      /// The arc observer registry.
+      typedef AlterationNotifier<AlterableGraphComponent, Edge> 
+      EdgeNotifier;
+      
+      /// \brief Gives back the arc alteration notifier.
+      ///
+      /// Gives back the arc alteration notifier.
+      EdgeNotifier& notifier(Edge) const {
+	return EdgeNotifier();
+      }
+
+      template <typename _Graph> 
+      struct Constraints {
+	void constraints() {
+	  checkConcept<AlterableGraphComponent<Base>, _Graph>();
+          typename _Graph::EdgeNotifier& uen 
+            = graph.notifier(typename _Graph::Edge());
+          ignore_unused_variable_warning(uen);
+	}
+	
+	const _Graph& graph;
+	
+      };
+      
+    };
+
+    /// \brief Class describing the concept of graph maps
+    /// 
+    /// This class describes the common interface of the graph maps
+    /// (NodeMap, ArcMap), that is \ref maps-page "maps" which can be used to
+    /// associate data to graph descriptors (nodes or arcs).
+    template <typename _Graph, typename _Item, typename _Value>
+    class GraphMap : public ReadWriteMap<_Item, _Value> {
+    public:
+
+      typedef ReadWriteMap<_Item, _Value> Parent;
+
+      /// The graph type of the map.
+      typedef _Graph Graph;
+      /// The key type of the map.
+      typedef _Item Key;
+      /// The value type of the map.
+      typedef _Value Value;
+
+      /// \brief Construct a new map.
+      ///
+      /// Construct a new map for the graph.
+      explicit GraphMap(const Graph&) {}
+      /// \brief Construct a new map with default value.
+      ///
+      /// Construct a new map for the graph and initalise the values.
+      GraphMap(const Graph&, const Value&) {}
+      /// \brief Copy constructor.
+      ///
+      /// Copy Constructor.
+      GraphMap(const GraphMap&) : Parent() {}
+      
+      /// \brief Assign operator.
+      ///
+      /// Assign operator. It does not mofify the underlying graph,
+      /// it just iterates on the current item set and set the  map
+      /// with the value returned by the assigned map. 
+      template <typename CMap>
+      GraphMap& operator=(const CMap&) { 
+        checkConcept<ReadMap<Key, Value>, CMap>();
+        return *this;
+      }
+
+      template<typename _Map>
+      struct Constraints {
+	void constraints() {
+	  checkConcept<ReadWriteMap<Key, Value>, _Map >();
+	  // Construction with a graph parameter
+	  _Map a(g);
+	  // Constructor with a graph and a default value parameter
+	  _Map a2(g,t);
+	  // Copy constructor.
+	  _Map b(c);
+          
+          ReadMap<Key, Value> cmap;
+          b = cmap;
+
+	  ignore_unused_variable_warning(a2);
+	  ignore_unused_variable_warning(b);
+	}
+
+	const _Map &c;
+	const Graph &g;
+	const typename GraphMap::Value &t;
+      };
+
+    };
+
+    /// \brief An empty mappable digraph class.
+    ///
+    /// This class provides beside the core digraph features
+    /// map interface for the digraph structure.
+    /// This concept is part of the Digraph concept.
+    template <typename _Base = BaseDigraphComponent>
+    class MappableDigraphComponent : public _Base  {
+    public:
+
+      typedef _Base Base;
+      typedef typename Base::Node Node;
+      typedef typename Base::Arc Arc;
+
+      typedef MappableDigraphComponent Digraph;
+
+      /// \brief ReadWrite map of the nodes.
+      ///
+      /// ReadWrite map of the nodes.
+      ///
+      template <typename _Value>
+      class NodeMap : public GraphMap<Digraph, Node, _Value> {
+      public:
+        typedef GraphMap<MappableDigraphComponent, Node, _Value> Parent;
+
+	/// \brief Construct a new map.
+	///
+	/// Construct a new map for the digraph.
+	explicit NodeMap(const MappableDigraphComponent& digraph) 
+          : Parent(digraph) {}
+
+	/// \brief Construct a new map with default value.
+	///
+	/// Construct a new map for the digraph and initalise the values.
+	NodeMap(const MappableDigraphComponent& digraph, const _Value& value)
+          : Parent(digraph, value) {}
+
+	/// \brief Copy constructor.
+	///
+	/// Copy Constructor.
+	NodeMap(const NodeMap& nm) : Parent(nm) {}
+
+	/// \brief Assign operator.
+	///
+	/// Assign operator.
+        template <typename CMap>
+        NodeMap& operator=(const CMap&) { 
+          checkConcept<ReadMap<Node, _Value>, CMap>();
+          return *this;
+        }
+
+      };
+
+      /// \brief ReadWrite map of the arcs.
+      ///
+      /// ReadWrite map of the arcs.
+      ///
+      template <typename _Value>
+      class ArcMap : public GraphMap<Digraph, Arc, _Value> {
+      public:
+        typedef GraphMap<MappableDigraphComponent, Arc, _Value> Parent;
+
+	/// \brief Construct a new map.
+	///
+	/// Construct a new map for the digraph.
+	explicit ArcMap(const MappableDigraphComponent& digraph) 
+          : Parent(digraph) {}
+
+	/// \brief Construct a new map with default value.
+	///
+	/// Construct a new map for the digraph and initalise the values.
+	ArcMap(const MappableDigraphComponent& digraph, const _Value& value)
+          : Parent(digraph, value) {}
+
+	/// \brief Copy constructor.
+	///
+	/// Copy Constructor.
+	ArcMap(const ArcMap& nm) : Parent(nm) {}
+
+	/// \brief Assign operator.
+	///
+	/// Assign operator.
+        template <typename CMap>
+        ArcMap& operator=(const CMap&) { 
+          checkConcept<ReadMap<Arc, _Value>, CMap>();
+          return *this;
+        }
+
+      };
+
+
+      template <typename _Digraph>
+      struct Constraints {
+
+	struct Dummy {
+	  int value;
+	  Dummy() : value(0) {}
+	  Dummy(int _v) : value(_v) {}
+	};
+
+	void constraints() {
+	  checkConcept<Base, _Digraph>();
+	  { // int map test
+	    typedef typename _Digraph::template NodeMap<int> IntNodeMap;
+	    checkConcept<GraphMap<_Digraph, typename _Digraph::Node, int>, 
+	      IntNodeMap >();
+	  } { // bool map test
+	    typedef typename _Digraph::template NodeMap<bool> BoolNodeMap;
+	    checkConcept<GraphMap<_Digraph, typename _Digraph::Node, bool>,
+	      BoolNodeMap >();
+	  } { // Dummy map test
+	    typedef typename _Digraph::template NodeMap<Dummy> DummyNodeMap;
+	    checkConcept<GraphMap<_Digraph, typename _Digraph::Node, Dummy>,
+	      DummyNodeMap >();
+	  } 
+
+	  { // int map test
+	    typedef typename _Digraph::template ArcMap<int> IntArcMap;
+	    checkConcept<GraphMap<_Digraph, typename _Digraph::Arc, int>,
+	      IntArcMap >();
+	  } { // bool map test
+	    typedef typename _Digraph::template ArcMap<bool> BoolArcMap;
+	    checkConcept<GraphMap<_Digraph, typename _Digraph::Arc, bool>,
+	      BoolArcMap >();
+	  } { // Dummy map test
+	    typedef typename _Digraph::template ArcMap<Dummy> DummyArcMap;
+	    checkConcept<GraphMap<_Digraph, typename _Digraph::Arc, Dummy>, 
+	      DummyArcMap >();
+	  } 
+	}
+
+	_Digraph& digraph;
+      };
+    };
+
+    /// \brief An empty mappable base bipartite 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 MappableDigraphComponent<_Base>  {
+    public:
+
+      typedef _Base Base;
+      typedef typename Base::Edge Edge;
+
+      typedef MappableGraphComponent Graph;
+
+      /// \brief ReadWrite map of the edges.
+      ///
+      /// ReadWrite map of the edges.
+      ///
+      template <typename _Value>
+      class EdgeMap : public GraphMap<Graph, Edge, _Value> {  
+      public:
+        typedef GraphMap<MappableGraphComponent, Edge, _Value> Parent;
+
+	/// \brief Construct a new map.
+	///
+	/// Construct a new map for the graph.
+	explicit EdgeMap(const MappableGraphComponent& graph) 
+          : Parent(graph) {}
+
+	/// \brief Construct a new map with default value.
+	///
+	/// Construct a new map for the graph and 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<MappableGraphComponent<Base>, _Graph>();
+
+	  { // int map test
+	    typedef typename _Graph::template EdgeMap<int> IntEdgeMap;
+	    checkConcept<GraphMap<_Graph, typename _Graph::Edge, int>,
+	      IntEdgeMap >();
+	  } { // bool map test
+	    typedef typename _Graph::template EdgeMap<bool> BoolEdgeMap;
+	    checkConcept<GraphMap<_Graph, typename _Graph::Edge, bool>,
+	      BoolEdgeMap >();
+	  } { // Dummy map test
+	    typedef typename _Graph::template EdgeMap<Dummy> DummyEdgeMap;
+	    checkConcept<GraphMap<_Graph, typename _Graph::Edge, Dummy>, 
+	      DummyEdgeMap >();
+	  } 
+	}
+
+	_Graph& graph;
+      };
+    };
+
+    /// \brief An empty extendable digraph class.
+    ///
+    /// This class provides beside the core digraph features digraph
+    /// extendable interface for the digraph structure.  The main
+    /// difference between the base and this interface is that the
+    /// digraph alterations should handled already on this level.
+    template <typename _Base = BaseDigraphComponent>
+    class ExtendableDigraphComponent : public _Base {
+    public:
+      typedef _Base Base;
+
+      typedef typename _Base::Node Node;
+      typedef typename _Base::Arc Arc;
+
+      /// \brief Adds a new node to the digraph.
+      ///
+      /// Adds a new node to the digraph.
+      ///
+      Node addNode() {
+	return INVALID;
+      }
+    
+      /// \brief Adds a new arc connects the given two nodes.
+      ///
+      /// Adds a new arc connects the the given two nodes.
+      Arc addArc(const Node&, const Node&) {
+	return INVALID;
+      }
+
+      template <typename _Digraph>
+      struct Constraints {
+	void constraints() {
+          checkConcept<Base, _Digraph>();
+	  typename _Digraph::Node node_a, node_b;
+	  node_a = digraph.addNode();
+	  node_b = digraph.addNode();
+	  typename _Digraph::Arc arc;
+	  arc = digraph.addArc(node_a, node_b);
+	}
+
+	_Digraph& digraph;
+      };
+    };
+
+    /// \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 = BaseGraphComponent>
+    class ExtendableGraphComponent : public _Base {
+    public:
+
+      typedef _Base Base;
+      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 arc connects the given two nodes.
+      ///
+      /// Adds a new arc connects the the given two nodes.
+      Edge addArc(const Node&, const Node&) {
+	return INVALID;
+      }
+
+      template <typename _Graph>
+      struct Constraints {
+	void constraints() {
+	  checkConcept<Base, _Graph>();
+	  typename _Graph::Node node_a, node_b;
+	  node_a = graph.addNode();
+	  node_b = graph.addNode();
+	  typename _Graph::Edge edge;
+	  edge = graph.addEdge(node_a, node_b);
+	}
+
+	_Graph& graph;
+      };
+    };
+
+    /// \brief An empty erasable digraph class.
+    ///  
+    /// This class provides beside the core digraph features core erase
+    /// functions for the digraph structure. The main difference between
+    /// the base and this interface is that the digraph alterations
+    /// should handled already on this level.
+    template <typename _Base = BaseDigraphComponent>
+    class ErasableDigraphComponent : public _Base {
+    public:
+
+      typedef _Base Base;
+      typedef typename Base::Node Node;
+      typedef typename Base::Arc Arc;
+
+      /// \brief Erase a node from the digraph.
+      ///
+      /// Erase a node from the digraph. This function should 
+      /// erase all arcs connecting to the node.
+      void erase(const Node&) {}    
+
+      /// \brief Erase an arc from the digraph.
+      ///
+      /// Erase an arc from the digraph.
+      ///
+      void erase(const Arc&) {}
+
+      template <typename _Digraph>
+      struct Constraints {
+	void constraints() {
+          checkConcept<Base, _Digraph>();
+	  typename _Digraph::Node node;
+	  digraph.erase(node);
+	  typename _Digraph::Arc arc;
+	  digraph.erase(arc);
+	}
+
+	_Digraph& digraph;
+      };
+    };
+
+    /// \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 = 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
+      /// arcs connecting to the node.
+      void erase(const Node&) {}    
+
+      /// \brief Erase an arc from the graph.
+      ///
+      /// Erase an arc from the graph.
+      ///
+      void erase(const Edge&) {}
+
+      template <typename _Graph>
+      struct Constraints {
+	void constraints() {
+          checkConcept<Base, _Graph>();
+	  typename _Graph::Node node;
+	  graph.erase(node);
+	  typename _Graph::Arc arc;
+	  graph.erase(arc);
+	}
+
+	_Graph& graph;
+      };
+    };
+
+    /// \brief An empty clearable base digraph class.
+    ///
+    /// This class provides beside the core digraph features core clear
+    /// functions for the digraph structure. The main difference between
+    /// the base and this interface is that the digraph alterations
+    /// should handled already on this level.
+    template <typename _Base = BaseDigraphComponent>
+    class ClearableDigraphComponent : public _Base {
+    public:
+
+      typedef _Base Base;
+
+      /// \brief Erase all nodes and arcs from the digraph.
+      ///
+      /// Erase all nodes and arcs from the digraph.
+      ///
+      void clear() {}    
+
+      template <typename _Digraph>
+      struct Constraints {
+	void constraints() {
+          checkConcept<Base, _Digraph>();
+	  digraph.clear();
+	}
+
+	_Digraph digraph;
+      };
+    };
+
+    /// \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 = BaseGraphComponent>
+    class ClearableGraphComponent : public ClearableDigraphComponent<_Base> {
+    public:
+
+      typedef _Base Base;
+
+      template <typename _Graph>
+      struct Constraints {
+	void constraints() {
+          checkConcept<ClearableGraphComponent<Base>, _Graph>();
+	}
+
+	_Graph graph;
+      };
+    };
+
+  }
+
+}
+
+#endif