/// Provides an immutable and unique id for each item in the graph.

   /// The IdMap class provides a unique and immutable id for each item of the

   /// same type (e.g. node) in the graph. This id is <ul><li>\b unique:

   /// different items (nodes) get different ids <li>\b immutable: the id of an

   /// item (node) does not change (even if you delete other nodes).  </ul>

   /// Through this map you get access (i.e. can read) the inner id values of

   /// the items stored in the graph. This map can be inverted with its member

   /// class \c InverseMap or with the \c operator() member.

   ///

   template <typename _Graph, typename _Item>

   class IdMap {

   public:

     typedef _Graph Graph;

     typedef int Value;

     typedef _Item Item;

     typedef _Item Key;

     /// \brief Constructor.

     ///

     /// Constructor of the map.

     explicit IdMap(const Graph& graph) : _graph(&graph) {}

     /// \brief Gives back the \e id of the item.

     ///

     /// Gives back the immutable and unique \e id of the item.

     int operator[](const Item& item) const { return _graph->id(item);}

     /// \brief Gives back the item by its id.

     ///

     /// Gives back the item by its id.

     Item operator()(int id) { return _graph->fromId(id, Item()); }

   private:

     const Graph* _graph;

   public:

     /// \brief The class represents the inverse of its owner (IdMap).

     ///

     /// The class represents the inverse of its owner (IdMap).

     /// \see inverse()

     class InverseMap {

     public:

       /// \brief Constructor.

       ///

       /// Constructor for creating an id-to-item map.

       explicit InverseMap(const Graph& graph) : _graph(&graph) {}

       /// \brief Constructor.

       ///

       /// Constructor for creating an id-to-item map.

       explicit InverseMap(const IdMap& map) : _graph(map._graph) {}

       /// \brief Gives back the given item from its id.

       ///

       /// Gives back the given item from its id.

       ///

       Item operator[](int id) const { return _graph->fromId(id, Item());}

     private:

       const Graph* _graph;

     };

     /// \brief Gives back the inverse of the map.

     ///

     /// Gives back the inverse of the IdMap.

     InverseMap inverse() const { return InverseMap(*_graph);}

   };

   /// \brief General invertable graph-map type.

   /// This type provides simple invertable graph-maps.

   /// The InvertableMap wraps an arbitrary ReadWriteMap

   /// and if a key is set to a new value then store it

   /// in the inverse map.

   ///

   /// The values of the map can be accessed

   /// with stl compatible forward iterator.

   ///

   /// \tparam _Graph The graph type.

   /// \tparam _Item The item type of the graph.

   /// \tparam _Value The value type of the map.

   ///

   /// \see IterableValueMap

   template <typename _Graph, typename _Item, typename _Value>

   class InvertableMap

     : protected ItemSetTraits<_Graph, _Item>::template Map<_Value>::Type {

   private:

     typedef typename ItemSetTraits<_Graph, _Item>::

     template Map<_Value>::Type Map;

     typedef _Graph Graph;

     typedef std::map<_Value, _Item> Container;

     Container _inv_map;

   public:

     /// The key type of InvertableMap (Node, Arc, Edge).

     typedef typename Map::Key Key;

     /// The value type of the InvertableMap.

     typedef typename Map::Value Value;

     /// \brief Constructor.

     ///

     /// Construct a new InvertableMap for the graph.

     ///

     explicit InvertableMap(const Graph& graph) : Map(graph) {}

     /// \brief Forward iterator for values.

     ///

     /// This iterator is an stl compatible forward

     /// iterator on the values of the map. The values can

     /// be accessed in the [beginValue, endValue) range.

     ///

     class ValueIterator

       : public std::iterator<std::forward_iterator_tag, Value> {

       friend class InvertableMap;

     private:

       ValueIterator(typename Container::const_iterator _it)

         : it(_it) {}

     public:

       ValueIterator() {}

       ValueIterator& operator++() { ++it; return *this; }

       ValueIterator operator++(int) {

         ValueIterator tmp(*this);

         operator++();

         return tmp;

       }

       const Value& operator*() const { return it->first; }

       const Value* operator->() const { return &(it->first); }

       bool operator==(ValueIterator jt) const { return it == jt.it; }

       bool operator!=(ValueIterator jt) const { return it != jt.it; }

     private:

       typename Container::const_iterator it;

     };

     /// \brief Returns an iterator to the first value.

     ///

     /// Returns an stl compatible iterator to the

     /// first value of the map. The values of the

     /// map can be accessed in the [beginValue, endValue)

     /// range.

     ValueIterator beginValue() const {

       return ValueIterator(_inv_map.begin());

     }

     /// \brief Returns an iterator after the last value.

     ///

     /// Returns an stl compatible iterator after the

     /// last value of the map. The values of the

     /// map can be accessed in the [beginValue, endValue)

     /// range.

     ValueIterator endValue() const {

       return ValueIterator(_inv_map.end());

     }

     /// \brief The setter function of the map.

     ///

     /// Sets the mapped value.

     void set(const Key& key, const Value& val) {

       Value oldval = Map::operator[](key);

       typename Container::iterator it = _inv_map.find(oldval);

       if (it != _inv_map.end() && it->second == key) {

         _inv_map.erase(it);

       }

       _inv_map.insert(make_pair(val, key));

       Map::set(key, val);

     }

     /// \brief The getter function of the map.

     ///

     /// It gives back the value associated with the key.

     typename MapTraits<Map>::ConstReturnValue

     operator[](const Key& key) const {

       return Map::operator[](key);

     }

     /// \brief Gives back the item by its value.

     ///

     /// Gives back the item by its value.

     Key operator()(const Value& key) const {

       typename Container::const_iterator it = _inv_map.find(key);

       return it != _inv_map.end() ? it->second : INVALID;

     }

   protected:

     /// \brief Erase the key from the map.

     ///

     /// Erase the key to the map. It is called by the

     /// \c AlterationNotifier.

     virtual void erase(const Key& key) {

       Value val = Map::operator[](key);

       typename Container::iterator it = _inv_map.find(val);

       if (it != _inv_map.end() && it->second == key) {

         _inv_map.erase(it);

       }

       Map::erase(key);

     }

     /// \brief Erase more keys from the map.

     ///

     /// Erase more keys from the map. It is called by the

     /// \c AlterationNotifier.

     virtual void erase(const std::vector<Key>& keys) {

       for (int i = 0; i < int(keys.size()); ++i) {

         Value val = Map::operator[](keys[i]);

         typename Container::iterator it = _inv_map.find(val);

         if (it != _inv_map.end() && it->second == keys[i]) {

           _inv_map.erase(it);

         }

       }

       Map::erase(keys);

     }

     /// \brief Clear the keys from the map and inverse map.

     ///

     /// Clear the keys from the map and inverse map. It is called by the

     /// \c AlterationNotifier.

     virtual void clear() {

       _inv_map.clear();

       Map::clear();

     }

   public:

     /// \brief The inverse map type.

     ///

     /// The inverse of this map. The subscript operator of the map

     /// gives back always the item what was last assigned to the value.

     class InverseMap {

     public:

       /// \brief Constructor of the InverseMap.

       ///

       /// Constructor of the InverseMap.

       explicit InverseMap(const InvertableMap& inverted)

         : _inverted(inverted) {}

       /// The value type of the InverseMap.

       typedef typename InvertableMap::Key Value;

       /// The key type of the InverseMap.

       typedef typename InvertableMap::Value Key;

       /// \brief Subscript operator.

       ///

       /// Subscript operator. It gives back always the item

       /// what was last assigned to the value.

       Value operator[](const Key& key) const {

         return _inverted(key);

       }

     private:

       const InvertableMap& _inverted;

     };

     /// \brief It gives back the just readable inverse map.

     ///

     /// It gives back the just readable inverse map.

     InverseMap inverse() const {

       return InverseMap(*this);

     }

   };

   /// \brief Provides a mutable, continuous and unique descriptor for each

   /// item in the graph.

   ///

   /// The DescriptorMap class provides a unique and continuous (but mutable)

   /// descriptor (id) for each item of the same type (e.g. node) in the

   /// graph. This id is <ul><li>\b unique: different items (nodes) get

   /// different ids <li>\b continuous: the range of the ids is the set of

   /// integers between 0 and \c n-1, where \c n is the number of the items of

   /// this type (e.g. nodes) (so the id of a node can change if you delete an

   /// other node, i.e. this id is mutable).  </ul> This map can be inverted

   /// with its member class \c InverseMap, or with the \c operator() member.

   ///

   /// \tparam _Graph The graph class the \c DescriptorMap belongs to.

   /// \tparam _Item The Item is the Key of the Map. It may be Node, Arc or

   /// Edge.

   template <typename _Graph, typename _Item>

   class DescriptorMap

     : protected ItemSetTraits<_Graph, _Item>::template Map<int>::Type {

     typedef _Item Item;

     typedef typename ItemSetTraits<_Graph, _Item>::template Map<int>::Type Map;

   public:

     /// The graph class of DescriptorMap.

     typedef _Graph Graph;

     /// The key type of DescriptorMap (Node, Arc, Edge).

     typedef typename Map::Key Key;

     /// The value type of DescriptorMap.

     typedef typename Map::Value Value;

     /// \brief Constructor.

     ///

     /// Constructor for descriptor map.

     explicit DescriptorMap(const Graph& _graph) : Map(_graph) {

       Item it;

       const typename Map::Notifier* nf = Map::notifier();

       for (nf->first(it); it != INVALID; nf->next(it)) {

         Map::set(it, _inv_map.size());

         _inv_map.push_back(it);

       }

     }

   protected:

     /// \brief Add a new key to the map.

     ///

     /// Add a new key to the map. It is called by the

     /// \c AlterationNotifier.

     virtual void add(const Item& item) {

       Map::add(item);

       Map::set(item, _inv_map.size());

       _inv_map.push_back(item);

     }

     /// \brief Add more new keys to the map.

     ///

     /// Add more new keys to the map. It is called by the

     /// \c AlterationNotifier.

     virtual void add(const std::vector<Item>& items) {

       Map::add(items);

       for (int i = 0; i < int(items.size()); ++i) {

         Map::set(items[i], _inv_map.size());

         _inv_map.push_back(items[i]);

       }

     }

     /// \brief Erase the key from the map.

     ///

     /// Erase the key from the map. It is called by the

     /// \c AlterationNotifier.

     virtual void erase(const Item& item) {

       Map::set(_inv_map.back(), Map::operator[](item));

       _inv_map[Map::operator[](item)] = _inv_map.back();

       _inv_map.pop_back();

       Map::erase(item);

     }

     /// \brief Erase more keys from the map.

     ///

     /// Erase more keys from the map. It is called by the

     /// \c AlterationNotifier.

     virtual void erase(const std::vector<Item>& items) {

       for (int i = 0; i < int(items.size()); ++i) {

         Map::set(_inv_map.back(), Map::operator[](items[i]));

         _inv_map[Map::operator[](items[i])] = _inv_map.back();

         _inv_map.pop_back();

       }

       Map::erase(items);

     }

     /// \brief Build the unique map.

     ///

     /// Build the unique map. It is called by the

     /// \c AlterationNotifier.

     virtual void build() {

       Map::build();

       Item it;

       const typename Map::Notifier* nf = Map::notifier();

       for (nf->first(it); it != INVALID; nf->next(it)) {

         Map::set(it, _inv_map.size());

         _inv_map.push_back(it);

       }

     }

     /// \brief Clear the keys from the map.

     ///

     /// Clear the keys from the map. It is called by the

     /// \c AlterationNotifier.

     virtual void clear() {

       _inv_map.clear();

       Map::clear();

     }

   public:

     /// \brief Returns the maximal value plus one.

     ///

     /// Returns the maximal value plus one in the map.

     unsigned int size() const {

       return _inv_map.size();

     }

     /// \brief Swaps the position of the two items in the map.

     ///

     /// Swaps the position of the two items in the map.

     void swap(const Item& p, const Item& q) {

       int pi = Map::operator[](p);

       int qi = Map::operator[](q);

       Map::set(p, qi);

       _inv_map[qi] = p;

       Map::set(q, pi);

       _inv_map[pi] = q;

     }

     /// \brief Gives back the \e descriptor of the item.

     ///

     /// Gives back the mutable and unique \e descriptor of the map.

     int operator[](const Item& item) const {

       return Map::operator[](item);

     }

     /// \brief Gives back the item by its descriptor.

     ///

     /// Gives back th item by its descriptor.

     Item operator()(int id) const {

       return _inv_map[id];

     }

   private:

     typedef std::vector<Item> Container;

     Container _inv_map;

   public:

     /// \brief The inverse map type of DescriptorMap.

     ///

     /// The inverse map type of DescriptorMap.

     class InverseMap {

     public:

       /// \brief Constructor of the InverseMap.

       ///

       /// Constructor of the InverseMap.

       explicit InverseMap(const DescriptorMap& inverted)

         : _inverted(inverted) {}

       /// The value type of the InverseMap.

       typedef typename DescriptorMap::Key Value;

       /// The key type of the InverseMap.

       typedef typename DescriptorMap::Value Key;

       /// \brief Subscript operator.

       ///

       /// Subscript operator. It gives back the item

       /// that the descriptor belongs to currently.

       Value operator[](const Key& key) const {

         return _inverted(key);

       }

       /// \brief Size of the map.

       ///

       /// Returns the size of the map.

       unsigned int size() const {

         return _inverted.size();

       }

     private:

       const DescriptorMap& _inverted;

     };

     /// \brief Gives back the inverse of the map.

     ///

     /// Gives back the inverse of the map.

     const InverseMap inverse() const {

       return InverseMap(*this);

     }

   };

   /// \brief Returns the source of the given arc.

   ///

   /// The SourceMap gives back the source Node of the given arc.

   /// \see TargetMap

   template <typename Digraph>

   class SourceMap {

   public:

     typedef typename Digraph::Node Value;

     typedef typename Digraph::Arc Key;

     /// \brief Constructor

     ///

     /// Constructor

     /// \param _digraph The digraph that the map belongs to.

     explicit SourceMap(const Digraph& digraph) : _digraph(digraph) {}

     /// \brief The subscript operator.

     ///

     /// The subscript operator.

     /// \param arc The arc

     /// \return The source of the arc

     Value operator[](const Key& arc) const {

       return _digraph.source(arc);

     }

   private:

     const Digraph& _digraph;

   };

   /// \brief Returns a \ref SourceMap class.

   ///

   /// This function just returns an \ref SourceMap class.

   /// \relates SourceMap

   template <typename Digraph>

   inline SourceMap<Digraph> sourceMap(const Digraph& digraph) {

     return SourceMap<Digraph>(digraph);

   }

   /// \brief Returns the target of the given arc.

   ///

   /// The TargetMap gives back the target Node of the given arc.

   /// \see SourceMap

   template <typename Digraph>

   class TargetMap {

   public:

     typedef typename Digraph::Node Value;

     typedef typename Digraph::Arc Key;

     /// \brief Constructor

     ///

     /// Constructor

     /// \param _digraph The digraph that the map belongs to.

     explicit TargetMap(const Digraph& digraph) : _digraph(digraph) {}

     /// \brief The subscript operator.

     ///

     /// The subscript operator.

     /// \param e The arc

     /// \return The target of the arc

     Value operator[](const Key& e) const {

       return _digraph.target(e);

     }

   private:

     const Digraph& _digraph;

   };

   /// \brief Returns a \ref TargetMap class.

   ///

   /// This function just returns a \ref TargetMap class.

   /// \relates TargetMap

   template <typename Digraph>

   inline TargetMap<Digraph> targetMap(const Digraph& digraph) {

     return TargetMap<Digraph>(digraph);

   }

   /// \brief Returns the "forward" directed arc view of an edge.

   ///

   /// Returns the "forward" directed arc view of an edge.

   /// \see BackwardMap

   template <typename Graph>

   class ForwardMap {

   public:

     typedef typename Graph::Arc Value;

     typedef typename Graph::Edge Key;

     /// \brief Constructor

     ///

     /// Constructor

     /// \param _graph The graph that the map belongs to.

     explicit ForwardMap(const Graph& graph) : _graph(graph) {}

     /// \brief The subscript operator.

     ///

     /// The subscript operator.

     /// \param key An edge

     /// \return The "forward" directed arc view of edge

     Value operator[](const Key& key) const {

       return _graph.direct(key, true);

     }

   private:

     const Graph& _graph;

   };

   /// \brief Returns a \ref ForwardMap class.

   ///

   /// This function just returns an \ref ForwardMap class.

   /// \relates ForwardMap

   template <typename Graph>

   inline ForwardMap<Graph> forwardMap(const Graph& graph) {

     return ForwardMap<Graph>(graph);

   }

   /// \brief Returns the "backward" directed arc view of an edge.

   ///

   /// Returns the "backward" directed arc view of an edge.

   /// \see ForwardMap

   template <typename Graph>

   class BackwardMap {

   public:

     typedef typename Graph::Arc Value;

     typedef typename Graph::Edge Key;

     /// \brief Constructor

     ///

     /// Constructor

     /// \param _graph The graph that the map belongs to.

     explicit BackwardMap(const Graph& graph) : _graph(graph) {}

     /// \brief The subscript operator.

     ///

     /// The subscript operator.

     /// \param key An edge

     /// \return The "backward" directed arc view of edge

     Value operator[](const Key& key) const {

       return _graph.direct(key, false);

     }

   private:

     const Graph& _graph;

   };

   /// \brief Returns a \ref BackwardMap class

   /// This function just returns a \ref BackwardMap class.

   /// \relates BackwardMap

   template <typename Graph>

   inline BackwardMap<Graph> backwardMap(const Graph& graph) {

     return BackwardMap<Graph>(graph);

   }

   /// \brief Potential difference map

   ///

   /// If there is an potential map on the nodes then we

   /// can get an arc map as we get the substraction of the

   /// values of the target and source.

   template <typename Digraph, typename NodeMap>

   class PotentialDifferenceMap {

   public:

     typedef typename Digraph::Arc Key;

     typedef typename NodeMap::Value Value;

     /// \brief Constructor

     ///

     /// Contructor of the map

     explicit PotentialDifferenceMap(const Digraph& digraph,

                                     const NodeMap& potential)

       : _digraph(digraph), _potential(potential) {}

     /// \brief Const subscription operator

     ///

     /// Const subscription operator

     Value operator[](const Key& arc) const {

       return _potential[_digraph.target(arc)] -

         _potential[_digraph.source(arc)];

     }

   private:

     const Digraph& _digraph;

     const NodeMap& _potential;

   };

   /// \brief Returns a PotentialDifferenceMap.

   ///

   /// This function just returns a PotentialDifferenceMap.

   /// \relates PotentialDifferenceMap

   template <typename Digraph, typename NodeMap>

   PotentialDifferenceMap<Digraph, NodeMap>

   potentialDifferenceMap(const Digraph& digraph, const NodeMap& potential) {

     return PotentialDifferenceMap<Digraph, NodeMap>(digraph, potential);

   }

   /// \brief Map of the node in-degrees.

   ///

   /// This map returns the in-degree of a node. Once it is constructed,

   /// the degrees are stored in a standard NodeMap, so each query is done

   /// in constant time. On the other hand, the values are updated automatically

   /// whenever the digraph changes.

   ///

   /// \warning Besides addNode() and addArc(), a digraph structure may provide

   /// alternative ways to modify the digraph. The correct behavior of InDegMap

   /// is not guarantied if these additional features are used. For example

   /// the functions \ref ListDigraph::changeSource() "changeSource()",

   /// \ref ListDigraph::changeTarget() "changeTarget()" and

   /// \ref ListDigraph::reverseArc() "reverseArc()"

   /// of \ref ListDigraph will \e not update the degree values correctly.

   ///

   /// \sa OutDegMap

   template <typename _Digraph>

   class InDegMap

     : protected ItemSetTraits<_Digraph, typename _Digraph::Arc>

       ::ItemNotifier::ObserverBase {

   public:

     typedef _Digraph Digraph;

     typedef int Value;

     typedef typename Digraph::Node Key;

     typedef typename ItemSetTraits<Digraph, typename Digraph::Arc>

     ::ItemNotifier::ObserverBase Parent;

   private:

     class AutoNodeMap

       : public ItemSetTraits<Digraph, Key>::template Map<int>::Type {

     public:

       typedef typename ItemSetTraits<Digraph, Key>::

       template Map<int>::Type Parent;

       AutoNodeMap(const Digraph& digraph) : Parent(digraph, 0) {}

       virtual void add(const Key& key) {

         Parent::add(key);

         Parent::set(key, 0);

       }

       virtual void add(const std::vector<Key>& keys) {

         Parent::add(keys);

         for (int i = 0; i < int(keys.size()); ++i) {

           Parent::set(keys[i], 0);

         }

       }

       virtual void build() {

         Parent::build();

         Key it;

         typename Parent::Notifier* nf = Parent::notifier();

         for (nf->first(it); it != INVALID; nf->next(it)) {

           Parent::set(it, 0);

         }

       }

     };

   public:

     /// \brief Constructor.

     ///

     /// Constructor for creating in-degree map.

     explicit InDegMap(const Digraph& digraph)

       : _digraph(digraph), _deg(digraph) {

       Parent::attach(_digraph.notifier(typename Digraph::Arc()));

       for(typename Digraph::NodeIt it(_digraph); it != INVALID; ++it) {

         _deg[it] = countInArcs(_digraph, it);

       }

     }

     /// Gives back the in-degree of a Node.

     int operator[](const Key& key) const {

       return _deg[key];

     }

   protected:

     typedef typename Digraph::Arc Arc;

     virtual void add(const Arc& arc) {

       ++_deg[_digraph.target(arc)];

     }

     virtual void add(const std::vector<Arc>& arcs) {

       for (int i = 0; i < int(arcs.size()); ++i) {

         ++_deg[_digraph.target(arcs[i])];

       }

     }

     virtual void erase(const Arc& arc) {

       --_deg[_digraph.target(arc)];

     }

     virtual void erase(const std::vector<Arc>& arcs) {

       for (int i = 0; i < int(arcs.size()); ++i) {

         --_deg[_digraph.target(arcs[i])];

       }

     }

     virtual void build() {

       for(typename Digraph::NodeIt it(_digraph); it != INVALID; ++it) {

         _deg[it] = countInArcs(_digraph, it);

       }

     }

     virtual void clear() {

       for(typename Digraph::NodeIt it(_digraph); it != INVALID; ++it) {

         _deg[it] = 0;

       }

     }

   private:

     const Digraph& _digraph;

     AutoNodeMap _deg;

   };

   /// \brief Map of the node out-degrees.

   ///

   /// This map returns the out-degree of a node. Once it is constructed,

   /// the degrees are stored in a standard NodeMap, so each query is done

   /// in constant time. On the other hand, the values are updated automatically

   /// whenever the digraph changes.

   ///

   /// \warning Besides addNode() and addArc(), a digraph structure may provide

   /// alternative ways to modify the digraph. The correct behavior of OutDegMap

   /// is not guarantied if these additional features are used. For example

   /// the functions \ref ListDigraph::changeSource() "changeSource()",

   /// \ref ListDigraph::changeTarget() "changeTarget()" and

   /// \ref ListDigraph::reverseArc() "reverseArc()"

   /// of \ref ListDigraph will \e not update the degree values correctly.

   ///

   /// \sa InDegMap

   template <typename _Digraph>

   class OutDegMap

     : protected ItemSetTraits<_Digraph, typename _Digraph::Arc>

       ::ItemNotifier::ObserverBase {

   public:

     typedef _Digraph Digraph;

     typedef int Value;

     typedef typename Digraph::Node Key;

     typedef typename ItemSetTraits<Digraph, typename Digraph::Arc>

     ::ItemNotifier::ObserverBase Parent;

   private:

     class AutoNodeMap

       : public ItemSetTraits<Digraph, Key>::template Map<int>::Type {

     public:

       typedef typename ItemSetTraits<Digraph, Key>::

       template Map<int>::Type Parent;

       AutoNodeMap(const Digraph& digraph) : Parent(digraph, 0) {}

       virtual void add(const Key& key) {

         Parent::add(key);

         Parent::set(key, 0);

       }

       virtual void add(const std::vector<Key>& keys) {

         Parent::add(keys);

         for (int i = 0; i < int(keys.size()); ++i) {

           Parent::set(keys[i], 0);

         }

       }

       virtual void build() {

         Parent::build();

         Key it;

         typename Parent::Notifier* nf = Parent::notifier();

         for (nf->first(it); it != INVALID; nf->next(it)) {

           Parent::set(it, 0);

         }

       }

     };

   public:

     /// \brief Constructor.

     ///

     /// Constructor for creating out-degree map.

     explicit OutDegMap(const Digraph& digraph)

       : _digraph(digraph), _deg(digraph) {

       Parent::attach(_digraph.notifier(typename Digraph::Arc()));

       for(typename Digraph::NodeIt it(_digraph); it != INVALID; ++it) {

         _deg[it] = countOutArcs(_digraph, it);

       }

     }

     /// Gives back the out-degree of a Node.

     int operator[](const Key& key) const {

       return _deg[key];

     }

   protected:

     typedef typename Digraph::Arc Arc;

     virtual void add(const Arc& arc) {

       ++_deg[_digraph.source(arc)];

     }

     virtual void add(const std::vector<Arc>& arcs) {

       for (int i = 0; i < int(arcs.size()); ++i) {

         ++_deg[_digraph.source(arcs[i])];

       }

     }

     virtual void erase(const Arc& arc) {

       --_deg[_digraph.source(arc)];

     }

     virtual void erase(const std::vector<Arc>& arcs) {

       for (int i = 0; i < int(arcs.size()); ++i) {

         --_deg[_digraph.source(arcs[i])];

       }

     }

     virtual void build() {

       for(typename Digraph::NodeIt it(_digraph); it != INVALID; ++it) {

         _deg[it] = countOutArcs(_digraph, it);

       }

     }

     virtual void clear() {

       for(typename Digraph::NodeIt it(_digraph); it != INVALID; ++it) {

         _deg[it] = 0;

       }

     }

   private:

     const Digraph& _digraph;

     AutoNodeMap _deg;

   };