Combinatorial Problems on $H$-graphs
Abstract
Bir\'{o}, Hujter, and Tuza introduced the concept of -graphs (1992), intersection graphs of connected subgraphs of a subdivision of a graph . 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 -graphs. We show that for any fixed containing a certain 3-node, 6-edge multigraph as a minor that the clique problem is APX-hard on -graphs and the isomorphism problem is isomorphism-complete. We also provide positive results on -graphs. Namely, when is a cactus the clique problem can be solved in polynomial time. Also, when a graph has a Helly -representation, the clique problem can be solved in polynomial time. Finally, we observe that one can use treewidth techniques to show that both the -clique and list -coloring problems are FPT on -graphs. These FPT results apply more generally to treewidth-bounded graph classes where treewidth is bounded by a function of the clique number.
Cite
@article{arxiv.1706.00575,
title = {Combinatorial Problems on $H$-graphs},
author = {Steven Chaplick and Peter Zeman},
journal= {arXiv preprint arXiv:1706.00575},
year = {2017}
}