Computing a clique tree with algorithm MLS (Maximal Label Search)

TL;DR

This paper extends the Maximal Label Search (MLS) algorithm to efficiently compute clique trees, perfect moplex orderings, and minimal separators in chordal graphs, and applies it to complement and non-chordal graphs.

Contribution

It introduces modifications to MLS for computing clique trees, pmo, and minimal separators, including a linear time algorithm for these computations.

Findings

MLS can be adapted to compute clique trees and pmo in linear time.

New conditions on labels enable detection of new cliques and separators.

The algorithm applies to both chordal and non-chordal graphs, including complements.

Abstract

Algorithm MLS (Maximal Label Search) is a graph search algorithm which generalizes algorithms MCS, LexBFS, LexDFS and MNS. On a chordal graph, MLS computes a peo (perfect elimination ordering) of the graph. We show how algorithm MLS can be modified to compute a pmo (perfect moplex ordering) as well as a clique tree and the minimal separators of a chordal graph. We give a necessary and sufficient condition on the labeling structure for the beginning of a new clique in the clique tree to be detected by a condition on labels. MLS is also used to compute a clique tree of the complement graph, and new cliques in the complement graph can be detected by a condition on labels for any labeling structure. A linear time algorithm computing a pmo and the generators of the maximal cliques and minimal separators w.r.t. this pmo of the complement graph is provided. On a non-chordal graph, algorithm MLSM is used to compute an atom tree of the clique minimal separator decomposition of any graph.