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Preprints 1993

Triangulating Graphs without Asteroidal Triples
Citation key Report-365-1993
Author Rolf H. Möhring
Pages 281-287
Year 1993
Journal Discrete Applied Mathematics, 1996
Volume 64
Number 365
Institution Technische Universität Berlin, Institut für Mathematik
Abstract An asteroidal triple\/ of a graph $G$ is a triple of mutually independent vertices such that, between any two of them, there exists a path that avoids the neighbourhood of the third. A triangulation of $G$ is a chordal graph $H$ on the same vertex set that contains $G$ as a subgraph, i. e., $V(G) = V(H)$ and $E(G) \subseteq E(H)$. We show that every ⊆-minimal triangulation of a graph $G$ without asteroidal triples is already an interval graph\/. This implies that the treewidth\/ of a graph $G$ without asteroidal triples equals its pathwidth\/. Another consequence is that the minimum number of additional edges in a triangulation of $G$ ( fill-in\/) equals the minimum number of edges needed to embed $G$ into an interval graph ( interval completion number\/). The proof is based on the interesting property that a minimal cover of all induced cycles of a graph $G$ without asteroidal triples by chords does not introduce new asteroidal triples. These results complement recent results by Corneil et al. about the linear structure of graphs without asteroidal triples.
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