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

  *

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

  *

  * Copyright (C) 2003-2009

  * 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.

  *

  */

 namespace lemon {

 /**

 [PAGE]sec_graph_structures[PAGE] Graph Structures

 The implementation of combinatorial algorithms heavily relies on

 efficient graph structures. Diverse applications require the

 usage of different physical graph storages.

 In \ref sec_basics, we have introduced a general digraph structure,

 \ref ListDigraph. Apart from this class, LEMON provides several

 other classes for handling directed and undirected graphs to meet the

 diverging requirements of the possible users. In order to save on running

 time or on memory usage, some structures may fail to support some graph

 features like node or arc/edge deletion.

 You are free to use the graph structure that fit your requirements the best,

 since most graph algorithms and auxiliary data structures can be used

 with any of them.

 [SEC]sec_graph_concepts[SEC] Graph Concepts

 In LEMON, there are various graph types, which are rather different, but

 they all conform to the corresponding \ref graph_concepts "graph concept",

 which defines the common part of the graph interfaces. 

 The \ref concepts::Digraph "Digraph concept" describes the common interface

 of directed graphs (without any sensible implementation), while

 the \ref concepts::Graph "Graph concept" describes the undirected graphs.

 Any generic graph algorithm should only exploit the features of the

 corresponding graph concept. (It should compile with the

 \ref concepts::Digraph "Digraph" or \ref concepts::Graph "Graph" type,

 but it will not run properly, of course.)

 The graph %concepts define the member classes for the iterators and maps

 along with some useful basic functions for obtaining the identifiers of

 the items, the end nodes of the arcs (or edges) and their iterators,

 etc. 

 An actual graph implementation may have various additional functionalities

 according to its purpose.

 [SEC]sec_digraph_types[SEC] Digraph Structures

 The already used \ref ListDigraph class is the most versatile directed

 graph structure. Apart from the general digraph functionalities, it

 provides operations for adding and removing nodes and arcs, changing

 the source or target node of an arc, and contracting and splitting nodes

 or arcs.

 \ref SmartDigraph is another general digraph implementation, which is

 significantly more efficient (both in terms of space and time), but it

 provides less functionality. For example, nodes and arcs cannot be

 removed from it. 

 \ref FullDigraph is an efficient implementation of a directed full graph.

 This structure is completely static, so you can neither add nor delete

 arcs or nodes, and the class needs constant space in memory.

 [SEC]sec_undir_graphs[SEC] Undirected Graphs

 LEMON also provides undirected graph structures. For example,

 \ref ListGraph and \ref SmartGraph are the undirected versions of

 \ref ListDigraph and \ref SmartDigraph, respectively.

 They provide similar features to the digraph structures.

 The \ref concepts::Graph "undirected graphs" also fulfill the concept of

 \ref concepts::Digraph "directed graphs", in such a way that each 

 undirected \e edge of a graph can also be regarded as two oppositely

 directed \e arcs. As a result, all directed graph algorithms automatically

 run on undirected graphs, as well.

 Undirected graphs provide an \c Edge type for the \e undirected \e edges

 and an \c Arc type for the \e directed \e arcs. The \c Arc type is

 convertible to \c Edge (or inherited from it), thus the corresponding

 edge can always be obtained from an arc.

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

 graph and the corresponding arcs are added or removed automatically

 (there are twice as many arcs as edges)

 For example,

 \code

   ListGraph g;

   ListGraph::Node a = g.addNode();

   ListGraph::Node b = g.addNode();

   ListGraph::Node c = g.addNode();

   ListGraph::Edge e = g.addEdge(a,b);

   g.addEdge(b,c);

   g.addEdge(c,a);

 \endcode

 Each edge has an inherent orientation, thus it can be defined whether an

 arc is forward or backward oriented in an undirected graph with respect

 to this default oriantation of the represented edge.

 The direction of an arc can be obtained and set using the functions

 \ref concepts::Graph::direction() "direction()" and

 \ref concepts::Graph::direct() "direct()", respectively.

 For example,

 \code

   ListGraph::Arc a1 = g.direct(e, true);    // a1 is the forward arc

   ListGraph::Arc a2 = g.direct(e, false);   // a2 is the backward arc

   if (a2 == g.oppositeArc(a1))

     std::cout << "a2 is the opposite of a1" << std::endl;

 \endcode

 The end nodes of an edge can be obtained using the functions

 \ref concepts::Graph::source() "u()" and

 \ref concepts::Graph::target() "v()", while the

 \ref concepts::Graph::source() "source()" and

 \ref concepts::Graph::target() "target()" can be used for arcs.

 \code

   std::cout << "Edge " << g.id(e) << " connects node "

     << g.id(g.u(e)) << " and node " << g.id(g.v(e)) << std::endl;

   std::cout << "Arc " << g.id(a2) << " goes from node "

     << g.id(g.source(a2)) << " to node " << g.id(g.target(a2)) << std::endl;

 \endcode

 Similarly to the digraphs, the undirected graphs also provide iterators

 \ref concepts::Graph::NodeIt "NodeIt", \ref concepts::Graph::ArcIt "ArcIt",

 \ref concepts::Graph::OutArcIt "OutArcIt" and \ref concepts::Graph::InArcIt

 "InArcIt", which can be used the same way.

 However, they also have iterator classes for edges.

 \ref concepts::Graph::EdgeIt "EdgeIt" traverses all edges in the graph and

 \ref concepts::Graph::IncEdgeIt "IncEdgeIt" lists the incident edges of a

 certain node.

 For example, the degree of each node can be computed and stored in a node map

 like this:

 \code

   ListGraph::NodeMap<int> deg(g, 0);

   for (ListGraph::NodeIt n(g); n != INVALID; ++n) {

     for (ListGraph::IncEdgeIt e(g, n); e != INVALID; ++e) {

       deg[n]++;

     }

   }

 \endcode

 In an undirected graph, both \ref concepts::Graph::OutArcIt "OutArcIt"

 and \ref concepts::Graph::InArcIt "InArcIt" iterates on the same \e edges

 but with opposite direction. They are convertible to both \c Arc and

 \c Edge types. \ref concepts::Graph::IncEdgeIt "IncEdgeIt" also iterates

 on these edges, but it is not convertible to \c Arc, only to \c Edge.

 Apart from the node and arc maps, an undirected graph also defines

 a template member class for constructing edge maps. These maps can be

 used in conjunction with both edges and arcs.

 For example,

 \code

   ListGraph::EdgeMap cost(g);

   cost[e] = 10;

   std::cout << cost[e] << std::endl;

   std::cout << cost[a1] << ", " << cost[a2] << std::endl;

   ListGraph::ArcMap arc_cost(g);

   arc_cost[a1] = cost[a1];

   arc_cost[a2] = 2 * cost[a2];

   // std::cout << arc_cost[e] << std::endl;   // this is not valid

   std::cout << arc_cost[a1] << ", " << arc_cost[a2] << std::endl;

 \endcode

 [SEC]sec_special_graphs[SEC] Special Graph Structures

 In addition to the general undirected classes \ref ListGraph and

 \ref SmartGraph, LEMON also provides special purpose graph types for

 handling \ref FullGraph "full graphs", \ref GridGraph "grid graphs" and

 \ref HypercubeGraph "hypercube graphs".

 They all static structures, i.e. they do not allow distinct item additions

 or deletions, the graph has to be built at once.

 [TRAILER]

 */

 }