# 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.

## 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.

## 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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.