Package org.jgrapht.alg.cycle

Class 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
    See Also:
    ChordalityInspector

    • Constructor Detail

      • 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 Detail

      • 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