};

   /// \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);

     }