Abstract
The algorithm MLS (Maximal Label Search) is a graph search algorithm that generalizes the algorithms Maximum Cardinality Search (MCS), Lexicographic Breadth-First Search (LexBFS), Lexicographic Depth-First Search (LexDFS) and Maximal Neighborhood Search (MNS). On a chordal graph, MLS computes a PEO (perfect elimination ordering) of the graph. We show how the 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 of MLS 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. We provide a linear time algorithm computing a PMO and the corresponding generators of the maximal cliques and minimal separators of the complement graph. On a non-chordal graph, the algorithm MLSM, a graph search algorithm computing an MEO and a minimal triangulation of the graph, is used to compute an atom tree of the clique minimal separator decomposition of any graph.
Highlights
Chordal graphs form an important and well-studied graph class, have many characterizations and properties and are used in many applications
We show how a clique tree of a chordal graph H can be computed from a perfect moplex ordering (PMO) of H by building the maximal cliques one after another (Corollary 1) and how the general algorithm MLS can be modified in order to compute a PMO, and a PMO, a clique tree and the minimal separators of any chordal graph for any labeling structure
The algorithm MLSM computes an minimal elimination ordering (MEO) and the associated minimal triangulation of the input graph G. It computes an minimal moplex ordering (MMO) of G if the order on labels is total [12]. It can be modified into the algorithm moplex-MLSM computing an MMO of G whether the order on labels is total or not, which can be extended to the algorithms MLSM-CliqueTree and DCL-MLSM-CliqueTree computing an MMO, the associated minimal triangulation H of G, a clique tree and the minimal separators of H
Summary
Chordal graphs form an important and well-studied graph class, have many characterizations and properties and are used in many applications. From an algorithmic point of view, connected chordal graphs are endowed with a compact representation as a clique tree, which organizes both the maximal cliques (which are the nodes of the tree) and the minimal separators (which label the edges): in a chordal graph, a minimal separator is the intersection of two maximal cliques, so each minimal separator is a clique (a characterization of chordal graphs [1]). Since a connected chordal graph has at most n maximal cliques, a clique tree has at most n nodes and less than n edges, a very efficient representation of the underlying chordal graph. A clique tree of a chordal graph can be computed efficiently using the characterization of a chordal graph as a graph, which has a PEO [2]. The algorithm MCS numbers the vertices using labels that count the number of Algorithms 2017, 10, 20; doi:10.3390/a10010020 www.mdpi.com/journal/algorithms
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