Weakly triangulated graphs

Generating weakly triangulated graphs

We show that a graph is weakly triangulated, or weakly chordal, if and only if it can be generated by starting with a graph with no edges, and repeatedly adding an edge, so that the new edge is not the middle edge of any chordless path with four vertices. This is a corollary of results due to Sritharan and Spinrad, and Hayward, Hoo ang and Maaray, and a natural analogue of a theorem due to Fulk...

Triangulated and Weakly Triangulated Graphs: Simpliciality in Vertices and Edges

Recognizing Weakly Triangulated Graphs by Edge Separability

We apply Lekkerkerker and Boland's recognition algorithm for tri-angulated graphs to the class of weakly triangulated graphs. This yields a new characterization of weakly triangulated graphs, as well as a new O(m 2) recognition algorithm which, unlike the previous ones, is not based on the notion of a 2-pair, but rather on the structural properties of the minimal separators of the graph. It als...

Trianguled and Weakly Triangulated Graphs: Simpliciality in Vertices and Edges

We introduce the notion of weak simpliciality, in order to extend to weakly triangulated graphs properties of triangulated graphs, using Hayward’s notion that a vertex in a triangulated graph behaves as an edge in a weakly triangulated graph. In particular, we use our definition of weak simplicial edge elimination ordering to bound the number of minimal separators to n+m, and derive an efficien...

Meyniel Weakly Triangulated Graphs - I: Co-perfect Orderability

We show that Meyniel weakly triangulated graphs are co-perfectly orderable (equivalently , that P 5-free weakly triangulated graphs are perfectly orderable). Our proof is algorithmic, and relies on a notion concerning separating sets, a property of weakly triangulated graphs, and several properties of Meyniel weakly triangulated graphs.