On some classes of quasi-triangulated graphs

A graph is quasi-triangulated if each of its induced subgraphs has a vertex that is either simplicial (its neighbours form a clique) or co-simplicial (its non-neighbours form an independent set) in the induced subgraph. The problem of characterizing quasi-triangulated graphs by minimal forbidden induced subgraphs is open. We show that a graph is quasi-triangulated if it does not contain as induced subgraph any of the graphs $C_5$, $P_5$, $P_3+P_3$, fence, or their complements. The fence is obtained from a $P_4$ by substituting two adjacent vertices for each end-vertex of the $P_4$ and two non-adjacent vertices for each interior vertex of the $P_4$. We also provide an efficient algorithm to recognize the subclass ${QT}_k$, for fixed $k\ge 0$, of quasi-triangulated graphs studied in the literature.