Abstract
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 recognizing H-graphs is Open image in new window -complete if H contains the diamond graph as a minor. On the other hand, for each tree T, we give a polynomial-time algorithm for recognizing T-graphs and an O(n4)-time algorithm for recognizing K1,d-graphs.
For the dominating set problem (parameterized by the size of H), we give Open image in new window - and Open image in new window -time algorithms on K1,d-graphs and H-graphs, respectively. Our dominating set algorithm for H-graphs also provides Open image in new window -time algorithms for the independent set and independent dominating set problems on H-graphs.