Connectivity and Other Graph Properties

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Detailed Description

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

Classes
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.

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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.

Parameters:

graph

The undirected graph.

Return values:

compMap

A 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.

Parameters:

digraph

The digraph.

Return values:

compMap

A 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:

digraph

The digraph.

Return values:

cutMap

A 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. any two edges are on same circle.

Returns:

true if the graph bi-node-connected.

Note:

By definition, the empty graph 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.

Parameters:

graph

The undirected graph.

Return values:

compMap

A 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:

graph

The undirected graph.

Return values:

cutMap

A 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.

Parameters:

graph

The undirected graph.

Return values:

compMap

A 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:

graph

The undirected graph.

Return values:

cutMap

A 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:

digraph

The digraph, which must be DAG.

Return values:

order

A 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:

digraph

The digraph.

Return values:

order

A 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.

Parameters:

graph

The undirected graph.

Return values:

partMap

A 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