 # OpenGraph DrawingFramework

v. 2022.02 (Dogwood)

Connectivity in Graphs and Digraphs

Provides functions dealing with connectivity in graphs and clustered. More...

## Classes

class  ogdf::ConnectivityTester
Naive implementation for testing the connectivity of a graph. More...

Calculate one or all Maximum Adjacency Ordering(s) of a given simple undirected graph. More...

class  ogdf::Triconnectivity
realizes Hopcroft/Tarjan algorithm for finding the triconnected components of a biconnected multi-graph More...

## Methods for clustered graphs

bool ogdf::isCConnected (const ClusterGraph &C)
Returns true iff cluster graph C is c-connected. More...

void ogdf::makeCConnected (ClusterGraph &C, Graph &G, List< edge > &addedEdges, bool simple=true)
Makes a cluster graph c-connected by adding edges. More...

## Methods for connectivity

bool ogdf::isConnected (const Graph &G)
Returns true iff G is connected. More...

void ogdf::makeConnected (Graph &G, List< edge > &added)
Makes G connected by adding a minimum number of edges. More...

void ogdf::makeConnected (Graph &G)
makes G connected by adding a minimum number of edges. More...

int ogdf::connectedComponents (const Graph &G, NodeArray< int > &component, List< node > *isolated=nullptr)
Computes the connected components of G and optionally generates a list of isolated nodes. More...

int ogdf::connectedComponents (const Graph &G)
Computes the amount of connected components of G. More...

bool ogdf::findCutVertices (const Graph &G, ArrayBuffer< node > &cutVertices, bool onlyOne=false)
Finds cut vertices and potential edges that could be added to turn the cut vertices into non-cut vertices. More...

bool ogdf::isBiconnected (const Graph &G, node &cutVertex)
Returns true iff G is biconnected. More...

bool ogdf::isBiconnected (const Graph &G)
Returns true iff G is biconnected. More...

void ogdf::makeBiconnected (Graph &G, List< edge > &added)
Makes G biconnected by adding edges. More...

void ogdf::makeBiconnected (Graph &G)
Makes G biconnected by adding edges. More...

int ogdf::biconnectedComponents (const Graph &G, EdgeArray< int > &component, int &nonEmptyComponents)
Computes the biconnected components of G. More...

int ogdf::biconnectedComponents (const Graph &G, EdgeArray< int > &component)
Computes the biconnected components of G. More...

bool ogdf::isTwoEdgeConnected (const Graph &graph)
Returns true iff graph is 2-edge-connected. More...

bool ogdf::isTriconnected (const Graph &G, node &s1, node &s2)
Returns true iff G is triconnected. More...

bool ogdf::isTriconnected (const Graph &G)
Returns true iff G is triconnected. More...

bool ogdf::isTriconnectedPrimitive (const Graph &G, node &s1, node &s2)
Returns true iff G is triconnected (using a quadratic time algorithm!). More...

bool ogdf::isTriconnectedPrimitive (const Graph &G)
Returns true iff G is triconnected (using a quadratic time algorithm!). More...

void ogdf::triangulate (Graph &G)
Triangulates a planarly embedded graph G by adding edges. More...

int ogdf::connectedIsolatedComponents (const Graph &G, List< node > &isolated, NodeArray< int > &component)
Computes the connected components of G and optionally generates a list of isolated nodes. More...

bool ogdf::findCutVertices (const Graph &G, ArrayBuffer< node > &cutVertices, ArrayBuffer< Tuple2< node, node >> &addEdges, bool onlyOne=false)
Finds cut vertices and potential edges that could be added to turn the cut vertices into non-cut vertices. More...

bool ogdf::isTwoEdgeConnected (const Graph &graph, edge &bridge)
Returns true iff graph is 2-edge-connected. More...

## Methods for directed graphs

int ogdf::strongComponents (const Graph &G, NodeArray< int > &component)
Computes the strongly connected components of the digraph G. More...

## Detailed Description

Provides functions dealing with connectivity in graphs and clustered.

## ◆ biconnectedComponents() [1/2]

 int ogdf::biconnectedComponents ( const Graph & G, EdgeArray< int > & component )
inline

Computes the biconnected components of G.

Assigns component numbers (0, 1, ...) to the edges of G. The component number of each edge is stored in the edge array component. Each self-loop is counted as one biconnected component and has its own component number.

Parameters
 G is the input graph. component is assigned a mapping from edges to component numbers.
Returns
the number of biconnected components (including self-loops) + the number of nodes without neighbours (that is, the number of nodes who have no incident edges or whose incident edges are all self-loops).

Definition at line 618 of file simple_graph_alg.h.

## ◆ biconnectedComponents() [2/2]

 int ogdf::biconnectedComponents ( const Graph & G, EdgeArray< int > & component, int & nonEmptyComponents )

Computes the biconnected components of G.

Assigns component numbers (0, 1, ...) to the edges of G. The component number of each edge is stored in the edge array component. Each self-loop is counted as one biconnected component and has its own component number.

Parameters
 G is the input graph. component is assigned a mapping from edges to component numbers.
Returns
the number of biconnected components (including self-loops) + the number of nodes without neighbours (that is, the number of nodes who have no incident edges or whose incident edges are all self-loops).
Parameters
 nonEmptyComponents is the number of non-empty components. The indices of component range from 0 to nonEmptyComponents - 1.

## ◆ connectedComponents() [1/2]

 int ogdf::connectedComponents ( const Graph & G )
inline

Computes the amount of connected components of G.

Parameters
 G is the input graph.
Returns
the amount of connected components.

Definition at line 502 of file simple_graph_alg.h.

## ◆ connectedComponents() [2/2]

 int ogdf::connectedComponents ( const Graph & G, NodeArray< int > & component, List< node > * isolated = nullptr )

Computes the connected components of G and optionally generates a list of isolated nodes.

Assigns component numbers (0, 1, ...) to the nodes of G. The component number of each node is stored in the node array component.

Parameters
 G is the input graph. component is assigned a mapping from nodes to component numbers. isolated is assigned the list of isolated nodes. An isolated node is a node without incident edges.
Returns
the number of connected components.

## ◆ connectedIsolatedComponents()

 int ogdf::connectedIsolatedComponents ( const Graph & G, List< node > & isolated, NodeArray< int > & component )
inline

Computes the connected components of G and optionally generates a list of isolated nodes.

Deprecated:

Assigns component numbers (0, 1, ...) to the nodes of G. The component number of each node is stored in the node array component.

Parameters
 G is the input graph. component is assigned a mapping from nodes to component numbers. isolated is assigned the list of isolated nodes. An isolated node is a node without incident edges.
Returns
the number of connected components.

Definition at line 512 of file simple_graph_alg.h.

## ◆ findCutVertices() [1/2]

 bool ogdf::findCutVertices ( const Graph & G, ArrayBuffer< node > & cutVertices, ArrayBuffer< Tuple2< node, node >> & addEdges, bool onlyOne = false )

Finds cut vertices and potential edges that could be added to turn the cut vertices into non-cut vertices.

Precondition
G must be connected.
Parameters
 G is the graph whose cut vertices should be found. cutVertices is assigned the cut vertices of the graph. onlyOne should be set to true if the search should stop after finding one cut vertex, to false if all cut vertices should be found.
Returns
true if the graph contains at least one cut vertex, false otherwise.
Parameters
 addEdges is assigned the tuples of nodes which have to be connected in order to turn each cut vertex into a non-cut vertex.

## ◆ findCutVertices() [2/2]

 bool ogdf::findCutVertices ( const Graph & G, ArrayBuffer< node > & cutVertices, bool onlyOne = false )
inline

Finds cut vertices and potential edges that could be added to turn the cut vertices into non-cut vertices.

Precondition
G must be connected.
Parameters
 G is the graph whose cut vertices should be found. cutVertices is assigned the cut vertices of the graph. onlyOne should be set to true if the search should stop after finding one cut vertex, to false if all cut vertices should be found.
Returns
true if the graph contains at least one cut vertex, false otherwise.

Definition at line 543 of file simple_graph_alg.h.

## ◆ isBiconnected() [1/2]

 bool ogdf::isBiconnected ( const Graph & G )
inline

Returns true iff G is biconnected.

Parameters
 G is the input graph.

Definition at line 568 of file simple_graph_alg.h.

## ◆ isBiconnected() [2/2]

 bool ogdf::isBiconnected ( const Graph & G, node & cutVertex )

Returns true iff G is biconnected.

Parameters
 G is the input graph. cutVertex If false is returned and G is connected, cutVertex is assigned a cut vertex in G, else it is assigned nullptr.

## ◆ isCConnected()

 bool ogdf::isCConnected ( const ClusterGraph & C )

Returns true iff cluster graph C is c-connected.

## ◆ isConnected()

 bool ogdf::isConnected ( const Graph & G )

Returns true iff G is connected.

Parameters
 G is the input graph.
Returns
true if G is connected, false otherwise.

## ◆ isTriconnected() [1/2]

 bool ogdf::isTriconnected ( const Graph & G )
inline

Returns true iff G is triconnected.

Parameters
 G is the input graph.
Returns
true if G is triconnected, false otherwise.

Definition at line 674 of file simple_graph_alg.h.

## ◆ isTriconnected() [2/2]

 bool ogdf::isTriconnected ( const Graph & G, node & s1, node & s2 )

Returns true iff G is triconnected.

If G is not triconnected then

• s1 and s2 are both nullptr if G is not connected.
• s1 is a cut vertex and s2 is nullptr if G is connected but not biconnected.
• s1 and s2 are a separation pair if G is bi- but not triconnected.
Parameters
 G is the input graph. s1 is assigned a cut vertex or one node of a separation pair, if G is not triconnected (see above). s2 is assigned one node of a separation pair, if G is not triconnected (see above).
Returns
true if G is triconnected, false otherwise.

## ◆ isTriconnectedPrimitive() [1/2]

 bool ogdf::isTriconnectedPrimitive ( const Graph & G )
inline

Returns true iff G is triconnected (using a quadratic time algorithm!).

Warning
This method has quadratic running time. An efficient linear time version is provided by isTriconnected().
Parameters
 G is the input graph.
Returns
true if G is triconnected, false otherwise.

Definition at line 710 of file simple_graph_alg.h.

## ◆ isTriconnectedPrimitive() [2/2]

 bool ogdf::isTriconnectedPrimitive ( const Graph & G, node & s1, node & s2 )

Returns true iff G is triconnected (using a quadratic time algorithm!).

If G is not triconnected then

• s1 and s2 are both nullptr if G is not connected.
• s1 is a cut vertex and s2 is nullptr if G is connected but not biconnected.
• s1 and s2 are a separation pair if G is bi- but not triconnected.
Warning
This method has quadratic running time. An efficient linear time version is provided by isTriconnected().
Parameters
 G is the input graph. s1 is assigned a cut vertex of one node of a separation pair, if G is not triconnected (see above). s2 is assigned one node of a separation pair, if G is not triconnected (see above).
Returns
true if G is triconnected, false otherwise.

## ◆ isTwoEdgeConnected() [1/2]

 bool ogdf::isTwoEdgeConnected ( const Graph & graph )
inline

Returns true iff graph is 2-edge-connected.

Implementation of the algorithm to determine 2-edge-connectivity from the following publication:

Jens M. Schmidt: A Simple Test on 2-Vertex- and 2-Edge-Connectivity. Information Processing Letters (2013)

It runs in O(|E|+|V|) as it relies on two DFS.

Parameters
 graph is the input graph.

Definition at line 644 of file simple_graph_alg.h.

## ◆ isTwoEdgeConnected() [2/2]

 bool ogdf::isTwoEdgeConnected ( const Graph & graph, edge & bridge )

Returns true iff graph is 2-edge-connected.

Implementation of the algorithm to determine 2-edge-connectivity from the following publication:

Jens M. Schmidt: A Simple Test on 2-Vertex- and 2-Edge-Connectivity. Information Processing Letters (2013)

It runs in O(|E|+|V|) as it relies on two DFS.

Parameters
 graph is the input graph. bridge If false is returned and graph is connected, bridge is assigned a bridge in graph, else it is assigned nullptr

## ◆ makeBiconnected() [1/2]

 void ogdf::makeBiconnected ( Graph & G )
inline

Makes G biconnected by adding edges.

Parameters
 G is the input graph.

Definition at line 590 of file simple_graph_alg.h.

## ◆ makeBiconnected() [2/2]

 void ogdf::makeBiconnected ( Graph & G, List< edge > & added )

Makes G biconnected by adding edges.

Parameters
 G is the input graph. added is assigned the list of inserted edges.

## ◆ makeCConnected()

 void ogdf::makeCConnected ( ClusterGraph & C, Graph & G, List< edge > & addedEdges, bool simple = true )

Makes a cluster graph c-connected by adding edges.

Parameters
 C is the input cluster graph. G is the graph associated with the cluster graph C; the function adds new edges to this graph. addedEdges is assigned the list of newly created edges. simple selects the method used: If set to true, a simple variant that does not guarantee to preserve planarity is used.

## ◆ makeConnected() [1/2]

 void ogdf::makeConnected ( Graph & G )
inline

makes G connected by adding a minimum number of edges.

Parameters
 G is the input graph.

Definition at line 470 of file simple_graph_alg.h.

## ◆ makeConnected() [2/2]

 void ogdf::makeConnected ( Graph & G, List< edge > & added )

Makes G connected by adding a minimum number of edges.

Parameters

## ◆ strongComponents()

 int ogdf::strongComponents ( const Graph & G, NodeArray< int > & component )

Computes the strongly connected components of the digraph G.

The function implements the algorithm by Tarjan.

Parameters
 G is the input graph. component is assigned a mapping from nodes to component numbers (0, 1, ...).
Returns
the number of strongly connected components.

## ◆ triangulate()

 void ogdf::triangulate ( Graph & G )

Triangulates a planarly embedded graph G by adding edges.

The result of this function is that G is made maximally planar by adding new edges. G will also be planarly embedded such that each face is a triangle.

Precondition
G is planar, simple and represents a combinatorial embedding (i.e. G is planarly embedded).
Parameters
 G is the input graph to which edges will be added.