Elsevier

Discrete Applied Mathematics

Volume 184, 31 March 2015, Pages 50-61
Discrete Applied Mathematics

Efficiently decomposing, recognizing and triangulating hole-free graphs without diamonds

https://doi.org/10.1016/j.dam.2014.11.018Get rights and content
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Abstract

A graph is hole- and diamond-free (HD-free) if none of its induced subgraphs is isomorphic to a chordless cycle of length at least five or to a diamond. Using the clique separator approach and the simple structure of atoms of HD-free graphs, we show how to recognize HD-free graphs in time O(n2). One of the main tools is Lexicographic Breadth-First Search (LexBFS); we give two new properties of LexBFS which are essential for reaching the time bound and which hold for any graph. Moreover, we find minimal triangulations of HD-free graphs in time O(n2), introducing efficient algorithms for the minimal triangulation of matched co-bipartite graphs and chordal bipartite graphs.

Keywords

Algorithms
Graph classes
Hole-free graphs
Diamond-free graphs
LexBFS
Clique separator decomposition
Recognition time bound
Minimal triangulations

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