Characterising Chordal Contact $$B_0$$-VPG Graphs

Characterising circular-arc contact B0–VPG graphs

“…We first introduce some known families of minimal forbidden induced subgraphs for the class of contact B 0 -VPG graphs. Let T [6] be the family of graphs containing H 0 (see Figure 1) as well as all graphs that can be partitioned into a nontrivial tree T of maximum degree at most three and the disjoint union of triangles, in such a way that each triangle is complete to a vertex v of T and anticomplete to T −{v}, every leaf v of T is complete to exactly two triangles, every vertex v of degree two in T is complete to exactly one triangle, and vertices of degree three in T have no neighbours outside T (see Figure 2).…”

“…Lemma 3.8. [6] Let G be a {K 5 , K 4 − e}-free graph. 1 If G is a minimal non contact B 0 -VPG graph, then every simplicial vertex of G has degree exactly three.…”

“…In order to recognise the class of contact B 0 -VPG graphs within circular-arc graphs, we first check whether the graph is chordal, which can be done in polynomial time [31]. If it is the case, we can apply the recognition algorithm of [6], whose output is either a contact B 0 -VPG representation or a forbidden induced subgraph. Otherwise, we obtain a hole in the graph, and we either find an induced K 4 − e in the graph or we can compute the structure of the graph with respect to this hole, as described in Corollary 3.6.…”

“…It was shown in [15,20] that recognising the class of contact B 0 -VPG graphs is NP-complete, and the complete list of minimal forbidden induced subgraphs for the class is not yet known. Nevertheless, characterisations of contact B 0 -VPG graphs by minimal forbidden induced subgraphs are known when restricted to some graph classes such as chordal, P 5 -free, P 4 -tidy, treecographs [6,7]; furthermore, most of those characterisations lead to polynomial-time recognition algorithms within the class. It is also known that every bipartite planar graph is contact B 0 -VPG [14].…”