/* -*- 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