Triangulating graphs without asteroidal triples

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) ⊂= 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. (1994) about the linear structure of graphs without asteroidal triples.