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Open AccessArticle

Computing a Clique Tree with the Algorithm Maximal Label Search

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LIMOS (Laboratoire d’Informatique, d’Optimisation et de Modélisation des Systèmes) UMR CNRS 6158, Ensemble Scientifique des Cézeaux, F-63178 Aubière CEDEX, France
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LIRMM (Laboratoire d’Informatique, de Robotique et de Microélectronique de Montpellier), 161 Rue Ada, F-34095 Montpellier CEDEX 5, France
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Author to whom correspondence should be addressed.
Academic Editor: Qianping Gu
Algorithms 2017, 10(1), 20; https://doi.org/10.3390/a10010020
Received: 29 October 2016 / Revised: 12 January 2017 / Accepted: 16 January 2017 / Published: 25 January 2017
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. View Full-Text
Keywords: chordal graph; clique tree; perfect elimination ordering; perfect moplex ordering; Maximal Label Search; LexBFS; MCS chordal graph; clique tree; perfect elimination ordering; perfect moplex ordering; Maximal Label Search; LexBFS; MCS
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Berry, A.; Simonet, G. Computing a Clique Tree with the Algorithm Maximal Label Search. Algorithms 2017, 10, 20.

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