On H-Topological Intersection Graphs

Abstract

Biró et al. (Discrete. Math 100(1–3):267–279, 1992) introduced the concept of H-graphs, intersection graphs of connected subgraphs of a subdivision of a graph H. They are related to and generalize many important classes of geometric intersection graphs, e.g., interval graphs, circular-arc graphs, split graphs, and chordal graphs. Our paper starts a new line of research in the area of geometric intersection graphs by studying several classical computational problems on H-graphs: recognition, graph isomorphism, dominating set, clique, and colorability. We negatively answer the 25-year-old question of Biró, Hujter, and Tuza which asks whether H-graphs can be recognized in polynomial time, for a fixed graph H. We prove that it is \(\textsf {NP}\)-complete if H contains the diamond graph as a minor. On the positive side, we provide a polynomial-time algorithm recognizing T-graphs, for each fixed tree T. For the special case when T is a star \(S_d\) of degree d, we have an \(\mathcal{O}(n^{3.5})\)-time algorithm. We give \(\textsf {FPT}\)- and \(\textsf {XP}\)-time algorithms solving the minimum dominating set problem on \(S_d\)-graphs and H-graphs, parametrized by d and the size of H, respectively. The algorithm for H-graphs adapts to an \(\textsf {XP}\)-time algorithm for the independent set and the independent dominating set problems on H-graphs. If H contains the double-triangle as a minor, we prove that the graph isomorphism problem is GI-complete and that the clique problem is APX-hard. On the positive side, we show that the clique problem can be solved in polynomial time if H is a cactus graph. Also, when a graph has a Helly H-representation, the clique problem is polynomial-time solvable. Further, we show that both the k-clique and the list k-coloring problems are solvable in FPT-time on H-graphs, parameterized by k and the treewidth of H. In fact, these results apply to classes of graphs with treewidth bounded by a function of the clique number. We observe that H-graphs have at most \(n^{O(\Vert H\Vert )}\) minimal separators which allows us to apply the meta-algorithmic framework of Fomin, Todinca, and Villanger (2015) to show that for each fixed t, finding a maximum induced subgraph of treewidth t can be done in polynomial time. In the case when H is a cactus, we improve the bound to \(O(\Vert H\Vert n^2)\).