New characterizations of Gallai’s i-triangulated graphs

Abstract

The i-triangulated graphs, introduced by Tibor Gallai in the early 1960s, have a number of characterizations—one of the simplest is that every odd cycle of length 5 or more has noncrossing chords. A variety of new characterizations are proved, starting with a simple forbidden induced subgraph characterization and including the following two:(1) Every 2-connected induced subgraph H is bipartite or chordal or, for every induced even cycle C of H, the intersection of the neighborhoods in H of all the vertices of C induces a complete multipartite subgraph.(2) For every 2-connected induced subgraph H that is not bipartite, every cardinality-2 minimal vertex separator of H induces an edge.