# Class ChordalGraphMinimalVertexSeparatorFinder<V,E>

java.lang.Object
org.jgrapht.alg.cycle.ChordalGraphMinimalVertexSeparatorFinder<V,E>
Type Parameters:
V - the graph vertex type
E - the graph edge type

public class ChordalGraphMinimalVertexSeparatorFinder<V,E> extends Object
Allows obtaining a mapping of all minimal vertex separators of a graph to their multiplicities

In the context of this implementation following definitions are used:

• A set of vertices $S$ of a graph $G=(V, E)$ is called a u-v separator, if vertices $u$ and $v$ in the induced graph on vertices $V(G) - S$ are in different connected components.
• A set $S$ is called a minimal u-v separator if it is a u-v separator and no proper subset of $S$ is a u-v separator.
• A set $S$ is called a minimal vertex separator if it is minimal u-v separator for some vertices $u$ and $v$ of the graph $G$.
• A set of vertices $S$ is called a minimal separator if no proper subset of $S$ is a separator of the graph $G$.

Let $\sigma = (v_1, v_2, \dots, v_n)$ be some perfect elimination order (peo) of the graph $G = (V, E)$. The induced graph on vertices $(v_1, v_2, \dots, v_i)$ with respect to peo $\sigma$ is denoted as $G_i$. The predecessors set of vertex $v$ with respect to peo $\sigma$ is denoted as $N(v, \sigma)$. A set $B$ is called a base set with respect to $\sigma$, is there exist some vertex $v$ with $t = \sigma(v)$ such that $N(v, \sigma) = B$ and B is not a maximal clique in $G_{t-1}$. The vertices which satisfy conditions described above are called dependent vertices with respect to $\sigma$. The cardinality of the set of dependent vertices is called a multiplicity of the base set $B$. The multiplicity of a minimal vertex separator indicates the number of different pairs of vertices separated by it.The definitions of a base set and a minimal vertex separator in the context of chordal graphs are equivalent.

For more information on the topic see: Kumar, P. Sreenivasa & Madhavan, C. E. Veni. (1998). Minimal vertex separators of chordal graphs. Discrete Applied Mathematics. 89. 155-168. 10.1016/S0166-218X(98)00123-1.

The running time of the algorithm is $\mathcal{O}(\omega(G)(|V| + |E|))$, where $\omega(G)$ is the size of a maximum clique of the graph $G$.

Author:
Timofey Chudakov
• ## Constructor Summary

Constructors
Constructor
Description
ChordalGraphMinimalVertexSeparatorFinder(Graph<V,E> graph)
Creates new ChordalGraphMinimalVertexSeparatorFinder instance.
• ## Method Summary

Modifier and Type
Method
Description
Set<Set<V>>
getMinimalSeparators()
Computes a set of all minimal separators of the graph and returns it.
Map<Set<V>,Integer>
getMinimalSeparatorsWithMultiplicities()
Computes a mapping of all minimal vertex separators of the graph and returns it.

### Methods inherited from class java.lang.Object

clone, equals, finalize, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait
• ## Constructor Details

• ### ChordalGraphMinimalVertexSeparatorFinder

public ChordalGraphMinimalVertexSeparatorFinder(Graph<V,E> graph)
Creates new ChordalGraphMinimalVertexSeparatorFinder instance. The ChordalityInspector used in this implementation uses the MaximumCardinalityIterator iterator
Parameters:
graph - the graph minimal separators to search in
• ## Method Details

• ### getMinimalSeparators

public Set<Set<V>> getMinimalSeparators()
Computes a set of all minimal separators of the graph and returns it. Returns null if the graph isn't chordal.
Returns:
computed set of all minimal separators, or null if the graph isn't chordal
• ### getMinimalSeparatorsWithMultiplicities

public Map<Set<V>,Integer> getMinimalSeparatorsWithMultiplicities()
Computes a mapping of all minimal vertex separators of the graph and returns it. Returns null if the graph isn't chordal.
Returns:
computed mapping of all minimal separators to their multiplicities, or null if the graph isn't chordal