Combinatorial Problems on H-graphs

Biró, Hujter, and Tuza introduced the concept of H-graphs (1992), intersection graphs of connected subgraphs of a subdivision of a graph H. They naturally generalize many important classes of graphs, e.g., interval graphs and circular-arc graphs. We continue the study of these graph classes by considering coloring, clique, and isomorphism problems on H-graphs. We show that for any fixed H containing a certain 3-node, 6-edge multigraph as a minor that the clique problem is APX-hard on H-graphs and the isomorphism problem is isomorphism-complete. We also provide positive results on H-graphs. Namely, when H is a cactus the clique problem can be solved in polynomial time. Also, when a graph G has a HellyH-representation, the clique problem can be solved in polynomial time. Finally, we observe that one can use treewidth techniques to show that both the k-clique and list k-coloring problems are FPT on H-graphs. These FPT results apply more generally to treewidth-bounded graph classes where treewidth is bounded by a function of the clique number.