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23 pages, 311 KB  
Article
Computing a Clique Tree with the Algorithm Maximal Label Search
by Anne Berry and Geneviève Simonet
Algorithms 2017, 10(1), 20; https://doi.org/10.3390/a10010020 - 25 Jan 2017
Cited by 4 | Viewed by 5959
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) [...] Read more.
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. Full article
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25 pages, 547 KB  
Article
Graph Extremities Defined by Search Algorithms
by Anne Berry, Jean R.S. Blair, Jean-Paul Bordat and Geneviève Simonet
Algorithms 2010, 3(2), 100-124; https://doi.org/10.3390/a3020100 - 24 Mar 2010
Cited by 21 | Viewed by 9720
Abstract
Graph search algorithms have exploited graph extremities, such as the leaves of a tree and the simplicial vertices of a chordal graph. Recently, several well-known graph search algorithms have been collectively expressed as two generic algorithms called MLS and MLSM. In this paper, [...] Read more.
Graph search algorithms have exploited graph extremities, such as the leaves of a tree and the simplicial vertices of a chordal graph. Recently, several well-known graph search algorithms have been collectively expressed as two generic algorithms called MLS and MLSM. In this paper, we investigate the properties of the vertex that is numbered 1 by MLS on a chordal graph and by MLSM on an arbitrary graph. We explain how this vertex is an extremity of the graph. Moreover, we show the remarkable property that the minimal separators included in the neighborhood of this vertex are totally ordered by inclusion. Full article
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