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Classes | Files | Functions
Connectivity and Other Graph Properties
Algorithms

Detailed Description

This group contains the algorithms for discovering the graph properties like connectivity, bipartiteness, euler property, simplicity etc.

connected_components.png

Classes

class  DiEulerIt< GR >
 Euler tour iterator for digraphs. More...
 
class  EulerIt< GR >
 Euler tour iterator for graphs. More...
 

Files

file  connectivity.h
 Connectivity algorithms.
 
file  euler.h
 Euler tour iterators and a function for checking the Eulerian property.
 

Functions

template<typename Graph >
bool connected (const Graph &graph)
 Check whether an undirected graph is connected.
 
template<typename Graph >
int countConnectedComponents (const Graph &graph)
 Count the number of connected components of an undirected graph.
 
template<class Graph , class NodeMap >
int connectedComponents (const Graph &graph, NodeMap &compMap)
 Find the connected components of an undirected graph.
 
template<typename Digraph >
bool stronglyConnected (const Digraph &digraph)
 Check whether a directed graph is strongly connected.
 
template<typename Digraph >
int countStronglyConnectedComponents (const Digraph &digraph)
 Count the number of strongly connected components of a directed graph.
 
template<typename Digraph , typename NodeMap >
int stronglyConnectedComponents (const Digraph &digraph, NodeMap &compMap)
 Find the strongly connected components of a directed graph.
 
template<typename Digraph , typename ArcMap >
int stronglyConnectedCutArcs (const Digraph &digraph, ArcMap &cutMap)
 Find the cut arcs of the strongly connected components.
 
template<typename Graph >
bool biNodeConnected (const Graph &graph)
 Check whether an undirected graph is bi-node-connected.
 
template<typename Graph >
int countBiNodeConnectedComponents (const Graph &graph)
 Count the number of bi-node-connected components of an undirected graph.
 
template<typename Graph , typename EdgeMap >
int biNodeConnectedComponents (const Graph &graph, EdgeMap &compMap)
 Find the bi-node-connected components of an undirected graph.
 
template<typename Graph , typename NodeMap >
int biNodeConnectedCutNodes (const Graph &graph, NodeMap &cutMap)
 Find the bi-node-connected cut nodes in an undirected graph.
 
template<typename Graph >
bool biEdgeConnected (const Graph &graph)
 Check whether an undirected graph is bi-edge-connected.
 
template<typename Graph >
int countBiEdgeConnectedComponents (const Graph &graph)
 Count the number of bi-edge-connected components of an undirected graph.
 
template<typename Graph , typename NodeMap >
int biEdgeConnectedComponents (const Graph &graph, NodeMap &compMap)
 Find the bi-edge-connected components of an undirected graph.
 
template<typename Graph , typename EdgeMap >
int biEdgeConnectedCutEdges (const Graph &graph, EdgeMap &cutMap)
 Find the bi-edge-connected cut edges in an undirected graph.
 
template<typename Digraph >
bool dag (const Digraph &digraph)
 Check whether a digraph is DAG.
 
template<typename Digraph , typename NodeMap >
void topologicalSort (const Digraph &digraph, NodeMap &order)
 Sort the nodes of a DAG into topolgical order.
 
template<typename Digraph , typename NodeMap >
bool checkedTopologicalSort (const Digraph &digraph, NodeMap &order)
 Sort the nodes of a DAG into topolgical order.
 
template<typename Graph >
bool acyclic (const Graph &graph)
 Check whether an undirected graph is acyclic.
 
template<typename Graph >
bool tree (const Graph &graph)
 Check whether an undirected graph is tree.
 
template<typename Graph >
bool bipartite (const Graph &graph)
 Check whether an undirected graph is bipartite.
 
template<typename Graph , typename NodeMap >
bool bipartitePartitions (const Graph &graph, NodeMap &partMap)
 Find the bipartite partitions of an undirected graph.
 
template<typename Graph >
bool loopFree (const Graph &graph)
 Check whether the given graph contains no loop arcs/edges.
 
template<typename Graph >
bool parallelFree (const Graph &graph)
 Check whether the given graph contains no parallel arcs/edges.
 
template<typename Graph >
bool simpleGraph (const Graph &graph)
 Check whether the given graph is simple.
 
template<typename GR >
bool eulerian (const GR &g)
 Check if the given graph is Eulerian.
 

Function Documentation

bool lemon::connected ( const Graph &  graph)

This function checks whether the given undirected graph is connected, i.e. there is a path between any two nodes in the graph.

Returns
true if the graph is connected.
Note
By definition, the empty graph is connected.
See Also
countConnectedComponents(), connectedComponents()
stronglyConnected()
int lemon::countConnectedComponents ( const Graph &  graph)

This function counts the number of connected components of the given undirected graph.

The connected components are the classes of an equivalence relation on the nodes of an undirected graph. Two nodes are in the same class if they are connected with a path.

Returns
The number of connected components.
Note
By definition, the empty graph consists of zero connected components.
See Also
connected(), connectedComponents()
int lemon::connectedComponents ( const Graph &  graph,
NodeMap &  compMap 
)

This function finds the connected components of the given undirected graph.

The connected components are the classes of an equivalence relation on the nodes of an undirected graph. Two nodes are in the same class if they are connected with a path.

connected_components.png
Parameters
graphThe undirected graph.
Return values
compMapA writable node map. The values will be set from 0 to the number of the connected components minus one. Each value of the map will be set exactly once, and the values of a certain component will be set continuously.
Returns
The number of connected components.
Note
By definition, the empty graph consists of zero connected components.
See Also
connected(), countConnectedComponents()
bool lemon::stronglyConnected ( const Digraph &  digraph)

This function checks whether the given directed graph is strongly connected, i.e. any two nodes of the digraph are connected with directed paths in both direction.

Returns
true if the digraph is strongly connected.
Note
By definition, the empty digraph is strongly connected.
See Also
countStronglyConnectedComponents(), stronglyConnectedComponents()
connected()
int lemon::countStronglyConnectedComponents ( const Digraph &  digraph)

This function counts the number of strongly connected components of the given directed graph.

The strongly connected components are the classes of an equivalence relation on the nodes of a digraph. Two nodes are in the same class if they are connected with directed paths in both direction.

Returns
The number of strongly connected components.
Note
By definition, the empty digraph has zero strongly connected components.
See Also
stronglyConnected(), stronglyConnectedComponents()
int lemon::stronglyConnectedComponents ( const Digraph &  digraph,
NodeMap &  compMap 
)

This function finds the strongly connected components of the given directed graph. In addition, the numbering of the components will satisfy that there is no arc going from a higher numbered component to a lower one (i.e. it provides a topological order of the components).

The strongly connected components are the classes of an equivalence relation on the nodes of a digraph. Two nodes are in the same class if they are connected with directed paths in both direction.

strongly_connected_components.png
Parameters
digraphThe digraph.
Return values
compMapA writable node map. The values will be set from 0 to the number of the strongly connected components minus one. Each value of the map will be set exactly once, and the values of a certain component will be set continuously.
Returns
The number of strongly connected components.
Note
By definition, the empty digraph has zero strongly connected components.
See Also
stronglyConnected(), countStronglyConnectedComponents()
int lemon::stronglyConnectedCutArcs ( const Digraph &  digraph,
ArcMap &  cutMap 
)

This function finds the cut arcs of the strongly connected components of the given digraph.

The strongly connected components are the classes of an equivalence relation on the nodes of a digraph. Two nodes are in the same class if they are connected with directed paths in both direction. The strongly connected components are separated by the cut arcs.

Parameters
digraphThe digraph.
Return values
cutMapA writable arc map. The values will be set to true for the cut arcs (exactly once for each cut arc), and will not be changed for other arcs.
Returns
The number of cut arcs.
See Also
stronglyConnected(), stronglyConnectedComponents()
bool lemon::biNodeConnected ( const Graph &  graph)

This function checks whether the given undirected graph is bi-node-connected, i.e. a connected graph without articulation node.

Returns
true if the graph bi-node-connected.
Note
By definition,
  • a graph consisting of zero or one node is bi-node-connected,
  • a graph consisting of two isolated nodes is not bi-node-connected and
  • a graph consisting of two nodes connected by an edge is bi-node-connected.
See Also
countBiNodeConnectedComponents(), biNodeConnectedComponents()
int countBiNodeConnectedComponents ( const Graph &  graph)

This function counts the number of bi-node-connected components of the given undirected graph.

The bi-node-connected components are the classes of an equivalence relation on the edges of a undirected graph. Two edges are in the same class if they are on same circle.

Returns
The number of bi-node-connected components.
See Also
biNodeConnected(), biNodeConnectedComponents()
int lemon::biNodeConnectedComponents ( const Graph &  graph,
EdgeMap &  compMap 
)

This function finds the bi-node-connected components of the given undirected graph.

The bi-node-connected components are the classes of an equivalence relation on the edges of a undirected graph. Two edges are in the same class if they are on same circle.

node_biconnected_components.png
Parameters
graphThe undirected graph.
Return values
compMapA writable edge map. The values will be set from 0 to the number of the bi-node-connected components minus one. Each value of the map will be set exactly once, and the values of a certain component will be set continuously.
Returns
The number of bi-node-connected components.
See Also
biNodeConnected(), countBiNodeConnectedComponents()
int lemon::biNodeConnectedCutNodes ( const Graph &  graph,
NodeMap &  cutMap 
)

This function finds the bi-node-connected cut nodes in the given undirected graph.

The bi-node-connected components are the classes of an equivalence relation on the edges of a undirected graph. Two edges are in the same class if they are on same circle. The bi-node-connected components are separted by the cut nodes of the components.

Parameters
graphThe undirected graph.
Return values
cutMapA writable node map. The values will be set to true for the nodes that separate two or more components (exactly once for each cut node), and will not be changed for other nodes.
Returns
The number of the cut nodes.
See Also
biNodeConnected(), biNodeConnectedComponents()
bool lemon::biEdgeConnected ( const Graph &  graph)

This function checks whether the given undirected graph is bi-edge-connected, i.e. any two nodes are connected with at least two edge-disjoint paths.

Returns
true if the graph is bi-edge-connected.
Note
By definition, the empty graph is bi-edge-connected.
See Also
countBiEdgeConnectedComponents(), biEdgeConnectedComponents()
int countBiEdgeConnectedComponents ( const Graph &  graph)

This function counts the number of bi-edge-connected components of the given undirected graph.

The bi-edge-connected components are the classes of an equivalence relation on the nodes of an undirected graph. Two nodes are in the same class if they are connected with at least two edge-disjoint paths.

Returns
The number of bi-edge-connected components.
See Also
biEdgeConnected(), biEdgeConnectedComponents()
int lemon::biEdgeConnectedComponents ( const Graph &  graph,
NodeMap &  compMap 
)

This function finds the bi-edge-connected components of the given undirected graph.

The bi-edge-connected components are the classes of an equivalence relation on the nodes of an undirected graph. Two nodes are in the same class if they are connected with at least two edge-disjoint paths.

edge_biconnected_components.png
Parameters
graphThe undirected graph.
Return values
compMapA writable node map. The values will be set from 0 to the number of the bi-edge-connected components minus one. Each value of the map will be set exactly once, and the values of a certain component will be set continuously.
Returns
The number of bi-edge-connected components.
See Also
biEdgeConnected(), countBiEdgeConnectedComponents()
int lemon::biEdgeConnectedCutEdges ( const Graph &  graph,
EdgeMap &  cutMap 
)

This function finds the bi-edge-connected cut edges in the given undirected graph.

The bi-edge-connected components are the classes of an equivalence relation on the nodes of an undirected graph. Two nodes are in the same class if they are connected with at least two edge-disjoint paths. The bi-edge-connected components are separted by the cut edges of the components.

Parameters
graphThe undirected graph.
Return values
cutMapA writable edge map. The values will be set to true for the cut edges (exactly once for each cut edge), and will not be changed for other edges.
Returns
The number of cut edges.
See Also
biEdgeConnected(), biEdgeConnectedComponents()
bool lemon::dag ( const Digraph &  digraph)

This function checks whether the given digraph is DAG, i.e. Directed Acyclic Graph.

Returns
true if there is no directed cycle in the digraph.
See Also
acyclic()
void lemon::topologicalSort ( const Digraph &  digraph,
NodeMap &  order 
)

This function sorts the nodes of the given acyclic digraph (DAG) into topolgical order.

Parameters
digraphThe digraph, which must be DAG.
Return values
orderA writable node map. The values will be set from 0 to the number of the nodes in the digraph minus one. Each value of the map will be set exactly once, and the values will be set descending order.
See Also
dag(), checkedTopologicalSort()
bool lemon::checkedTopologicalSort ( const Digraph &  digraph,
NodeMap &  order 
)

This function sorts the nodes of the given acyclic digraph (DAG) into topolgical order and also checks whether the given digraph is DAG.

Parameters
digraphThe digraph.
Return values
orderA readable and writable node map. The values will be set from 0 to the number of the nodes in the digraph minus one. Each value of the map will be set exactly once, and the values will be set descending order.
Returns
false if the digraph is not DAG.
See Also
dag(), topologicalSort()
bool lemon::acyclic ( const Graph &  graph)

This function checks whether the given undirected graph is acyclic.

Returns
true if there is no cycle in the graph.
See Also
dag()
bool lemon::tree ( const Graph &  graph)

This function checks whether the given undirected graph is tree.

Returns
true if the graph is acyclic and connected.
See Also
acyclic(), connected()
bool lemon::bipartite ( const Graph &  graph)

The function checks whether the given undirected graph is bipartite.

Returns
true if the graph is bipartite.
See Also
bipartitePartitions()
bool lemon::bipartitePartitions ( const Graph &  graph,
NodeMap &  partMap 
)

This function checks whether the given undirected graph is bipartite and gives back the bipartite partitions.

bipartite_partitions.png
Parameters
graphThe undirected graph.
Return values
partMapA writable node map of bool (or convertible) value type. The values will be set to true for one component and false for the other one.
Returns
true if the graph is bipartite, false otherwise.
See Also
bipartite()
bool lemon::loopFree ( const Graph &  graph)

This function returns true if there are no loop arcs/edges in the given graph. It works for both directed and undirected graphs.

bool lemon::parallelFree ( const Graph &  graph)

This function returns true if there are no parallel arcs/edges in the given graph. It works for both directed and undirected graphs.

bool lemon::simpleGraph ( const Graph &  graph)

This function returns true if the given graph is simple, i.e. it contains no loop arcs/edges and no parallel arcs/edges. The function works for both directed and undirected graphs.

See Also
loopFree(), parallelFree()
bool lemon::eulerian ( const GR &  g)

This function checks if the given graph is Eulerian. It works for both directed and undirected graphs.

By definition, a digraph is called Eulerian if and only if it is connected and the number of incoming and outgoing arcs are the same for each node. Similarly, an undirected graph is called Eulerian if and only if it is connected and the number of incident edges is even for each node.

Note
There are (di)graphs that are not Eulerian, but still have an Euler tour, since they may contain isolated nodes.
See Also
DiEulerIt, EulerIt