On H-Topological Intersection 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. Our paper is the first study of the recognition and dominating set problems of this large collection of intersection classes of graphs. We negatively answer the question of Biró, Hujter, and Tuza who asked whether H-graphs can be recognized in polynomial time, for a fixed graph H. Namely, we show that when H is the diamond graph, the recognition problem is NP-complete. However, for each tree T , we give a polynomial-time algorithm for recognizing T -graphs and an O(n4)-time algorithm for recognizing star-graphs, i.e., when T isK1,t for some t. For the dominating set problem (parameterized by the size of H), we give FPTand XP-time algorithms on star-graphs and H-graphs, respectively. Our dominating set algorithm for H-graphs also provides XPtime algorithms for the independent set and independent dominating set problems on H-graphs (again parameterized by ‖H‖). 1998 ACM Subject Classification G.2.2 Graph Theory, I.1.2 Algorithms