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

  *

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

  *

  * Copyright (C) 2003-2013

  * 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 undirected graphs.

 #ifndef LEMON_CONCEPTS_GRAPH_H

 #define LEMON_CONCEPTS_GRAPH_H

 #include <lemon/concepts/graph_components.h>

 #include <lemon/concepts/maps.h>

 #include <lemon/concept_check.h>

 #include <lemon/core.h>

 #include <lemon/bits/stl_iterators.h>

 namespace lemon {

   namespace concepts {

     /// \ingroup graph_concepts

     ///

     /// \brief Class describing the concept of undirected graphs.

     ///

     /// This class describes the common interface of all undirected

     /// graphs.

     ///

     /// Like all concept classes, it only provides an interface

     /// without any sensible implementation. So any general algorithm for

     /// undirected graphs should compile with this class, but it will not

     /// run properly, of course.

     /// An actual graph implementation like \ref ListGraph or

     /// \ref SmartGraph may have additional functionality.

     ///

     /// The undirected graphs also fulfill the concept of \ref Digraph

     /// "directed graphs", since each edge can also be regarded as two

     /// oppositely directed arcs.

     /// Undirected graphs provide an Edge type for the undirected edges and

     /// an Arc type for the directed arcs. The Arc type is convertible to

     /// Edge or inherited from it, i.e. the corresponding edge can be

     /// obtained from an arc.

     /// EdgeIt and EdgeMap classes can be used for the edges, while ArcIt

     /// and ArcMap classes can be used for the arcs (just like in digraphs).

     /// Both InArcIt and OutArcIt iterates on the same edges but with

     /// opposite direction. IncEdgeIt also iterates on the same edges

     /// as OutArcIt and InArcIt, but it is not convertible to Arc,

     /// only to Edge.

     ///

     /// In LEMON, each undirected edge has an inherent orientation.

     /// Thus it can defined if an arc is forward or backward oriented in

     /// an undirected graph with respect to this default oriantation of

     /// the represented edge.

     /// With the direction() and direct() functions the direction

     /// of an arc can be obtained and set, respectively.

     ///

     /// Only nodes and edges can be added to or removed from an undirected

     /// graph and the corresponding arcs are added or removed automatically.

     ///

     /// \sa Digraph

     class Graph {

     private:

       /// Graphs are \e not copy constructible. Use GraphCopy instead.

       Graph(const Graph&) {}

       /// \brief Assignment of a graph to another one is \e not allowed.

       /// Use GraphCopy instead.

       void operator=(const Graph&) {}

     public:

       /// Default constructor.

       Graph() {}

       /// \brief Undirected graphs should be tagged with \c UndirectedTag.

       ///

       /// Undirected graphs should be tagged with \c UndirectedTag.

       ///

       /// This tag helps the \c enable_if technics to make compile time

       /// specializations for undirected graphs.

       typedef True UndirectedTag;

       /// The node type of the graph

       /// This class identifies a node of the graph. It also serves

       /// as a base class of the node iterators,

       /// thus they convert to this type.

       class Node {

       public:

         /// Default constructor

         /// Default constructor.

         /// \warning It sets the object to an undefined value.

         Node() { }

         /// Copy constructor.

         /// Copy constructor.

         ///

         Node(const Node&) { }

         /// %Invalid constructor \& conversion.

         /// Initializes the object to be invalid.

         /// \sa Invalid for more details.

         Node(Invalid) { }

         /// Equality operator

         /// Equality operator.

         ///

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

         /// same object or both are \c INVALID.

         bool operator==(Node) const { return true; }

         /// Inequality operator

         /// Inequality operator.

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

         /// Artificial ordering operator.

         /// Artificial ordering operator.

         ///

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

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

         /// ordering of the items.

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

       };

       /// Iterator class for the nodes.

       /// This iterator goes through each node of the graph.

       /// Its usage is quite simple, for example, you can count the number

       /// of nodes in a graph \c g of type \c %Graph like this:

       ///\code

       /// int count=0;

       /// for (Graph::NodeIt n(g); n!=INVALID; ++n) ++count;

       ///\endcode

       class NodeIt : public Node {

       public:

         /// Default constructor

         /// Default constructor.

         /// \warning It sets the iterator to an undefined value.

         NodeIt() { }

         /// Copy constructor.

         /// Copy constructor.

         ///

         NodeIt(const NodeIt& n) : Node(n) { }

         /// %Invalid constructor \& conversion.

         /// Initializes the iterator to be invalid.

         /// \sa Invalid for more details.

         NodeIt(Invalid) { }

         /// Sets the iterator to the first node.

         /// Sets the iterator to the first node of the given digraph.

         ///

         explicit NodeIt(const Graph&) { }

         /// Sets the iterator to the given node.

         /// Sets the iterator to the given node of the given digraph.

         ///

         NodeIt(const Graph&, const Node&) { }

         /// Next node.

         /// Assign the iterator to the next node.

         ///

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

       };

       /// \brief Gets the collection of the nodes of the graph.

       ///

       /// This function can be used for iterating on

       /// the nodes of the graph. It returns a wrapped NodeIt, which looks

       /// like an STL container (by having begin() and end())

       /// which you can use in range-based for loops, STL algorithms, etc.

       /// For example you can write:

       ///\code

       /// ListGraph g;

       /// for(auto v: g.nodes())

       ///   doSomething(v);

       ///

       /// //Using an STL algorithm:

       /// copy(g.nodes().begin(), g.nodes().end(), vect.begin());

       ///\endcode

       LemonRangeWrapper1<NodeIt, Graph> nodes() const {

         return LemonRangeWrapper1<NodeIt, Graph>(*this);

       }

       /// The edge type of the graph

       /// This class identifies an edge of the graph. It also serves

       /// as a base class of the edge iterators,

       /// thus they will convert to this type.

       class Edge {

       public:

         /// Default constructor

         /// Default constructor.

         /// \warning It sets the object to an undefined value.

         Edge() { }

         /// Copy constructor.

         /// Copy constructor.

         ///

         Edge(const Edge&) { }

         /// %Invalid constructor \& conversion.

         /// Initializes the object to be invalid.

         /// \sa Invalid for more details.

         Edge(Invalid) { }

         /// Equality operator

         /// Equality operator.

         ///

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

         /// same object or both are \c INVALID.

         bool operator==(Edge) const { return true; }

         /// Inequality operator

         /// Inequality operator.

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

         /// Artificial ordering operator.

         /// Artificial ordering operator.

         ///

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

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

         /// ordering of the edges.

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

       };

       /// Iterator class for the edges.

       /// This iterator goes through each edge of the graph.

       /// Its usage is quite simple, for example, you can count the number

       /// of edges in a graph \c g of type \c %Graph as follows:

       ///\code

       /// int count=0;

       /// for(Graph::EdgeIt e(g); e!=INVALID; ++e) ++count;

       ///\endcode

       class EdgeIt : public Edge {

       public:

         /// Default constructor

         /// Default constructor.

         /// \warning It sets the iterator to an undefined value.

         EdgeIt() { }

         /// Copy constructor.

         /// Copy constructor.

         ///

         EdgeIt(const EdgeIt& e) : Edge(e) { }

         /// %Invalid constructor \& conversion.

         /// Initializes the iterator to be invalid.

         /// \sa Invalid for more details.

         EdgeIt(Invalid) { }

         /// Sets the iterator to the first edge.

         /// Sets the iterator to the first edge of the given graph.

         ///

         explicit EdgeIt(const Graph&) { }

         /// Sets the iterator to the given edge.

         /// Sets the iterator to the given edge of the given graph.

         ///

         EdgeIt(const Graph&, const Edge&) { }

         /// Next edge

         /// Assign the iterator to the next edge.

         ///

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

       };

       /// \brief Gets the collection of the edges of the graph.

       ///

       /// This function can be used for iterating on the

       /// edges of the graph. It returns a wrapped

       /// EdgeIt, which looks like an STL container

       /// (by having begin() and end()) which you can use in range-based

       /// for loops, STL algorithms, etc.

       /// For example you can write:

       ///\code

       /// ListGraph g;

       /// for(auto e: g.edges())

       ///   doSomething(e);

       ///

       /// //Using an STL algorithm:

       /// copy(g.edges().begin(), g.edges().end(), vect.begin());

       ///\endcode

       LemonRangeWrapper1<EdgeIt, Graph> edges() const {

         return LemonRangeWrapper1<EdgeIt, Graph>(*this);

       }

       /// Iterator class for the incident edges of a node.

       /// This iterator goes trough the incident undirected edges

       /// of a certain node of a graph.

       /// Its usage is quite simple, for example, you can compute the

       /// degree (i.e. the number of incident edges) of a node \c n

       /// in a graph \c g of type \c %Graph as follows.

       ///

       ///\code

       /// int count=0;

       /// for(Graph::IncEdgeIt e(g, n); e!=INVALID; ++e) ++count;

       ///\endcode

       ///

       /// \warning Loop edges will be iterated twice.

       class IncEdgeIt : public Edge {

       public:

         /// Default constructor

         /// Default constructor.

         /// \warning It sets the iterator to an undefined value.

         IncEdgeIt() { }

         /// Copy constructor.

         /// Copy constructor.

         ///

         IncEdgeIt(const IncEdgeIt& e) : Edge(e) { }

         /// %Invalid constructor \& conversion.

         /// Initializes the iterator to be invalid.

         /// \sa Invalid for more details.

         IncEdgeIt(Invalid) { }

         /// Sets the iterator to the first incident edge.

         /// Sets the iterator to the first incident edge of the given node.

         ///

         IncEdgeIt(const Graph&, const Node&) { }

         /// Sets the iterator to the given edge.

         /// Sets the iterator to the given edge of the given graph.

         ///

         IncEdgeIt(const Graph&, const Edge&) { }

         /// Next incident edge

         /// Assign the iterator to the next incident edge

         /// of the corresponding node.

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

       };

       /// \brief Gets the collection of the incident undirected edges

       ///  of a certain node of the graph.

       ///

       /// This function can be used for iterating on the

       /// incident undirected edges of a certain node of the graph.

       /// It returns a wrapped

       /// IncEdgeIt, which looks like an STL container

       /// (by having begin() and end()) which you can use in range-based

       /// for loops, STL algorithms, etc.

       /// For example if g is a Graph and u is a Node, you can write:

       ///\code

       /// for(auto e: g.incEdges(u))

       ///   doSomething(e);

       ///

       /// //Using an STL algorithm:

       /// copy(g.incEdges(u).begin(), g.incEdges(u).end(), vect.begin());

       ///\endcode

       LemonRangeWrapper2<IncEdgeIt, Graph, Node> incEdges(const Node& u) const {

         return LemonRangeWrapper2<IncEdgeIt, Graph, Node>(*this, u);

       }

       /// The arc type of the graph

       /// This class identifies a directed arc of the graph. It also serves

       /// as a base class of the arc iterators,

       /// thus they will convert to this type.

       class Arc {

       public:

         /// Default constructor

         /// Default constructor.

         /// \warning It sets the object to an undefined value.

         Arc() { }

         /// Copy constructor.

         /// Copy constructor.

         ///

         Arc(const Arc&) { }

         /// %Invalid constructor \& conversion.

         /// Initializes the object to be invalid.

         /// \sa Invalid for more details.

         Arc(Invalid) { }

         /// Equality operator

         /// Equality operator.

         ///

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

         /// same object or both are \c INVALID.

         bool operator==(Arc) const { return true; }

         /// Inequality operator

         /// Inequality operator.

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

         /// Artificial ordering operator.

         /// Artificial ordering operator.

         ///

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

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

         /// ordering of the arcs.

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

         /// Converison to \c Edge

         /// Converison to \c Edge.

         ///

         operator Edge() const { return Edge(); }

       };

       /// Iterator class for the arcs.

       /// This iterator goes through each directed arc of the graph.

       /// Its usage is quite simple, for example, you can count the number

       /// of arcs in a graph \c g of type \c %Graph as follows:

       ///\code

       /// int count=0;

       /// for(Graph::ArcIt a(g); a!=INVALID; ++a) ++count;

       ///\endcode

       class ArcIt : public Arc {

       public:

         /// Default constructor

         /// Default constructor.

         /// \warning It sets the iterator to an undefined value.

         ArcIt() { }

         /// Copy constructor.

         /// Copy constructor.

         ///

         ArcIt(const ArcIt& e) : Arc(e) { }

         /// %Invalid constructor \& conversion.

         /// Initializes the iterator to be invalid.

         /// \sa Invalid for more details.

         ArcIt(Invalid) { }

         /// Sets the iterator to the first arc.

         /// Sets the iterator to the first arc of the given graph.

         ///

         explicit ArcIt(const Graph &g) {

           ::lemon::ignore_unused_variable_warning(g);

         }

         /// Sets the iterator to the given arc.

         /// Sets the iterator to the given arc of the given graph.

         ///

         ArcIt(const Graph&, const Arc&) { }

         /// Next arc

         /// Assign the iterator to the next arc.

         ///

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

       };

       /// \brief Gets the collection of the directed arcs of the graph.

       ///

       /// This function can be used for iterating on the

       /// arcs of the graph. It returns a wrapped

       /// ArcIt, which looks like an STL container

       /// (by having begin() and end()) which you can use in range-based

       /// for loops, STL algorithms, etc.

       /// For example you can write:

       ///\code

       /// ListGraph g;

       /// for(auto a: g.arcs())

       ///   doSomething(a);

       ///

       /// //Using an STL algorithm:

       /// copy(g.arcs().begin(), g.arcs().end(), vect.begin());

       ///\endcode

       LemonRangeWrapper1<ArcIt, Graph> arcs() const {

         return LemonRangeWrapper1<ArcIt, Graph>(*this);

       }

       /// Iterator class for the outgoing arcs of a node.

       /// This iterator goes trough the \e outgoing directed arcs of a

       /// certain node of a graph.

       /// Its usage is quite simple, for example, you can count the number

       /// of outgoing arcs of a node \c n

       /// in a graph \c g of type \c %Graph as follows.

       ///\code

       /// int count=0;

       /// for (Digraph::OutArcIt a(g, n); a!=INVALID; ++a) ++count;

       ///\endcode

       class OutArcIt : public Arc {

       public:

         /// Default constructor

         /// Default constructor.

         /// \warning It sets the iterator to an undefined value.

         OutArcIt() { }

         /// Copy constructor.

         /// Copy constructor.

         ///

         OutArcIt(const OutArcIt& e) : Arc(e) { }

         /// %Invalid constructor \& conversion.

         /// Initializes the iterator to be invalid.

         /// \sa Invalid for more details.

         OutArcIt(Invalid) { }

         /// Sets the iterator to the first outgoing arc.

         /// Sets the iterator to the first outgoing arc of the given node.

         ///

         OutArcIt(const Graph& n, const Node& g) {

           ::lemon::ignore_unused_variable_warning(n);

           ::lemon::ignore_unused_variable_warning(g);

         }

         /// Sets the iterator to the given arc.

         /// Sets the iterator to the given arc of the given graph.

         ///

         OutArcIt(const Graph&, const Arc&) { }

         /// Next outgoing arc

         /// Assign the iterator to the next

         /// outgoing arc of the corresponding node.

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

       };

       /// \brief Gets the collection of the outgoing directed arcs of a

       /// certain node of the graph.

       ///

       /// This function can be used for iterating on the

       /// outgoing arcs of a certain node of the graph. It returns a wrapped

       /// OutArcIt, which looks like an STL container

       /// (by having begin() and end()) which you can use in range-based

       /// for loops, STL algorithms, etc.

       /// For example if g is a Graph and u is a Node, you can write:

       ///\code

       /// for(auto a: g.outArcs(u))

       ///   doSomething(a);

       ///

       /// //Using an STL algorithm:

       /// copy(g.outArcs(u).begin(), g.outArcs(u).end(), vect.begin());

       ///\endcode

       LemonRangeWrapper2<OutArcIt, Graph, Node> outArcs(const Node& u) const {

         return LemonRangeWrapper2<OutArcIt, Graph, Node>(*this, u);

       }

       /// Iterator class for the incoming arcs of a node.

       /// This iterator goes trough the \e incoming directed arcs of a

       /// certain node of a graph.

       /// Its usage is quite simple, for example, you can count the number

       /// of incoming arcs of a node \c n

       /// in a graph \c g of type \c %Graph as follows.

       ///\code

       /// int count=0;

       /// for (Digraph::InArcIt a(g, n); a!=INVALID; ++a) ++count;

       ///\endcode

       class InArcIt : public Arc {

       public:

         /// Default constructor

         /// Default constructor.

         /// \warning It sets the iterator to an undefined value.

         InArcIt() { }

         /// Copy constructor.

         /// Copy constructor.

         ///

         InArcIt(const InArcIt& e) : Arc(e) { }

         /// %Invalid constructor \& conversion.

         /// Initializes the iterator to be invalid.

         /// \sa Invalid for more details.

         InArcIt(Invalid) { }

         /// Sets the iterator to the first incoming arc.

         /// Sets the iterator to the first incoming arc of the given node.

         ///

         InArcIt(const Graph& g, const Node& n) {

           ::lemon::ignore_unused_variable_warning(n);

           ::lemon::ignore_unused_variable_warning(g);

         }

         /// Sets the iterator to the given arc.

         /// Sets the iterator to the given arc of the given graph.

         ///

         InArcIt(const Graph&, const Arc&) { }

         /// Next incoming arc

         /// Assign the iterator to the next

         /// incoming arc of the corresponding node.

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

       };

       /// \brief Gets the collection of the incoming directed arcs of

       /// a certain node of the graph.

       ///

       /// This function can be used for iterating on the

       /// incoming directed arcs of a certain node of the graph. It returns

       /// a wrapped InArcIt, which looks like an STL container

       /// (by having begin() and end()) which you can use in range-based

       /// for loops, STL algorithms, etc.

       /// For example if g is a Graph and u is a Node, you can write:

       ///\code

       /// for(auto a: g.inArcs(u))

       ///   doSomething(a);

       ///

       /// //Using an STL algorithm:

       /// copy(g.inArcs(u).begin(), g.inArcs(u).end(), vect.begin());

       ///\endcode

       LemonRangeWrapper2<InArcIt, Graph, Node> inArcs(const Node& u) const {

         return LemonRangeWrapper2<InArcIt, Graph, Node>(*this, u);

       }

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

       ///

       /// Standard graph map type for the nodes.

       /// It conforms to the ReferenceMap concept.

       template<class T>

       class NodeMap : public ReferenceMap<Node, T, T&, const T&>

       {

       public:

         /// Constructor

         explicit NodeMap(const Graph&) { }

         /// Constructor with given initial value

         NodeMap(const Graph&, T) { }

       private:

         ///Copy constructor

         NodeMap(const NodeMap& nm) :

           ReferenceMap<Node, T, T&, const T&>(nm) { }

         ///Assignment operator

         template <typename CMap>

         NodeMap& operator=(const CMap&) {

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

           return *this;

         }

       };

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

       ///

       /// Standard graph map type for the arcs.

       /// It conforms to the ReferenceMap concept.

       template<class T>

       class ArcMap : public ReferenceMap<Arc, T, T&, const T&>

       {

       public:

         /// Constructor

         explicit ArcMap(const Graph&) { }

         /// Constructor with given initial value

         ArcMap(const Graph&, T) { }

       private:

         ///Copy constructor

         ArcMap(const ArcMap& em) :

           ReferenceMap<Arc, T, T&, const T&>(em) { }

         ///Assignment operator

         template <typename CMap>

         ArcMap& operator=(const CMap&) {

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

           return *this;

         }

       };

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

       ///

       /// Standard graph map type for the edges.

       /// It conforms to the ReferenceMap concept.

       template<class T>

       class EdgeMap : public ReferenceMap<Edge, T, T&, const T&>

       {

       public:

         /// Constructor

         explicit EdgeMap(const Graph&) { }

         /// Constructor with given initial value

         EdgeMap(const Graph&, T) { }

       private:

         ///Copy constructor

         EdgeMap(const EdgeMap& em) :

           ReferenceMap<Edge, T, T&, const T&>(em) {}

         ///Assignment operator

         template <typename CMap>

         EdgeMap& operator=(const CMap&) {

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

           return *this;

         }

       };

       /// \brief The first node of the edge.

       ///

       /// Returns the first node of the given edge.

       ///

       /// Edges don't have source and target nodes, however, methods

       /// u() and v() are used to query the two end-nodes of an edge.

       /// The orientation of an edge that arises this way is called

       /// the inherent direction, it is used to define the default

       /// direction for the corresponding arcs.

       /// \sa v()

       /// \sa direction()

       Node u(Edge) const { return INVALID; }

       /// \brief The second node of the edge.

       ///

       /// Returns the second node of the given edge.

       ///

       /// Edges don't have source and target nodes, however, methods

       /// u() and v() are used to query the two end-nodes of an edge.

       /// The orientation of an edge that arises this way is called

       /// the inherent direction, it is used to define the default

       /// direction for the corresponding arcs.

       /// \sa u()

       /// \sa direction()

       Node v(Edge) const { return INVALID; }

       /// \brief The source node of the arc.

       ///

       /// Returns the source node of the given arc.

       Node source(Arc) const { return INVALID; }

       /// \brief The target node of the arc.

       ///

       /// Returns the target node of the given arc.

       Node target(Arc) const { return INVALID; }

       /// \brief The ID of the node.

       ///

       /// Returns the ID of the given node.

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

       /// \brief The ID of the edge.

       ///

       /// Returns the ID of the given edge.

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

       /// \brief The ID of the arc.

       ///

       /// Returns the ID of the given arc.

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

       /// \brief The node with the given ID.

       ///

       /// Returns the node with the given ID.

       /// \pre The argument should be a valid node ID in the graph.

       Node nodeFromId(int) const { return INVALID; }

       /// \brief The edge with the given ID.

       ///

       /// Returns the edge with the given ID.

       /// \pre The argument should be a valid edge ID in the graph.

       Edge edgeFromId(int) const { return INVALID; }

       /// \brief The arc with the given ID.

       ///

       /// Returns the arc with the given ID.

       /// \pre The argument should be a valid arc ID in the graph.

       Arc arcFromId(int) const { return INVALID; }

       /// \brief An upper bound on the node IDs.

       ///

       /// Returns an upper bound on the node IDs.

       int maxNodeId() const { return -1; }

       /// \brief An upper bound on the edge IDs.

       ///

       /// Returns an upper bound on the edge IDs.

       int maxEdgeId() const { return -1; }

       /// \brief An upper bound on the arc IDs.

       ///

       /// Returns an upper bound on the arc IDs.

       int maxArcId() const { return -1; }

       /// \brief The direction of the arc.

       ///

       /// Returns \c true if the direction of the given arc is the same as

       /// the inherent orientation of the represented edge.

       bool direction(Arc) const { return true; }

       /// \brief Direct the edge.

       ///

       /// Direct the given edge. The returned arc

       /// represents the given edge and its direction comes

       /// from the bool parameter. If it is \c true, then the direction

       /// of the arc is the same as the inherent orientation of the edge.

       Arc direct(Edge, bool) const {

         return INVALID;

       }

       /// \brief Direct the edge.

       ///

       /// Direct the given edge. The returned arc represents the given

       /// edge and its source node is the given node.

       Arc direct(Edge, Node) const {

         return INVALID;

       }

       /// \brief The oppositely directed arc.

       ///

       /// Returns the oppositely directed arc representing the same edge.

       Arc oppositeArc(Arc) const { return INVALID; }

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

       ///

       /// Returns the opposite node on the given edge.

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

       void first(Node&) const {}

       void next(Node&) const {}

       void first(Edge&) const {}

       void next(Edge&) const {}

       void first(Arc&) const {}

       void next(Arc&) const {}

       void firstOut(Arc&, Node) const {}

       void nextOut(Arc&) const {}

       void firstIn(Arc&, Node) const {}

       void nextIn(Arc&) const {}

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

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

       // The second parameter is dummy.

       Node fromId(int, Node) const { return INVALID; }

       // The second parameter is dummy.

       Edge fromId(int, Edge) const { return INVALID; }

       // The second parameter is dummy.

       Arc fromId(int, Arc) const { return INVALID; }

       // Dummy parameter.

       int maxId(Node) const { return -1; }

       // Dummy parameter.

       int maxId(Edge) const { return -1; }

       // Dummy parameter.

       int maxId(Arc) const { return -1; }

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

       ///

       /// Returns the base node of the given incident edge iterator.

       Node baseNode(IncEdgeIt) const { return INVALID; }

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

       ///

       /// Returns the running node of the given incident edge iterator.

       Node runningNode(IncEdgeIt) const { return INVALID; }

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

       ///

       /// Returns the base node of the given outgoing arc iterator

       /// (i.e. the source node of the corresponding arc).

       Node baseNode(OutArcIt) const { return INVALID; }

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

       ///

       /// Returns the running node of the given outgoing arc iterator

       /// (i.e. the target node of the corresponding arc).

       Node runningNode(OutArcIt) const { return INVALID; }

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

       ///

       /// Returns the base node of the given incoming arc iterator

       /// (i.e. the target node of the corresponding arc).

       Node baseNode(InArcIt) const { return INVALID; }

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

       ///

       /// Returns the running node of the given incoming arc iterator

       /// (i.e. the source node of the corresponding arc).

       Node runningNode(InArcIt) const { return INVALID; }

       template <typename _Graph>

       struct Constraints {

         void constraints() {

           checkConcept<BaseGraphComponent, _Graph>();

           checkConcept<IterableGraphComponent<>, _Graph>();

           checkConcept<IDableGraphComponent<>, _Graph>();

           checkConcept<MappableGraphComponent<>, _Graph>();

         }

       };

     };

   }

 }

 #endif