On H-topological intersection graphs
Peter Zeman


Abstract
Bir\'{o}, 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 Bir\'{o}, 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 $\cNP$-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 $\calO(n^4)$-time algorithm for recognizing $K_{1,d}$-graphs.  For the dominating set problem (parameterized by the size of $H$), we give $\cFPT$- and $\cXP$-time algorithms on $K_{1,d}$-graphs and $H$-graphs, respectively.  Our dominating set algorithm for $H$-graphs also provides $\cXP$-time algorithms for the independent set and independent dominating set problems on $H$-graphs.