Module org.jgrapht.core
Package org.jgrapht.alg.cycle

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 $G$.

See Also:

  • Field Details

    • graph

      private final Graph<V,E> graph
      The graph in which minimal vertex separators to searched in

    • chordalityInspector

      private final ChordalityInspector<V,E> chordalityInspector
      ChordalityInspector for testing chordality of the graph

    • minimalSeparatorsWithMultiplicities

      private Map<Set<V>,Integer> minimalSeparatorsWithMultiplicities
      A mapping of minimal separators to their multiplicities

  • 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

    • lazyComputeMinimalSeparatorsWithMultiplicities

      private void lazyComputeMinimalSeparatorsWithMultiplicities()
      Lazy computes a set of all minimal separators and a mapping of all minimal vertex separators to their multiplicities

    • getVertexInOrder

      private Map<V,Integer> getVertexInOrder(List<V> vertexOrder)
      Returns a map containing vertices from the vertexOrder mapped to their indices in vertexOrder.
      Parameters:
      vertexOrder - a list with vertices.
      Returns:
      a mapping of vertices from vertexOrder to their indices in vertexOrder.

    • getPredecessors

      private Set<V> getPredecessors(Map<V,Integer> vertexInOrder, V vertex)
      Returns the predecessors of vertex in the order defined by map. More precisely, returns those of vertex, whose mapped index in map is less then the index of vertex.
      Parameters:
      vertexInOrder - defines the mapping of vertices in graph to their indices in order.
      vertex - the vertex whose predecessors in order are to be returned.
      Returns:
      the predecessors of vertex in order defines by map.