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- W4300451657 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. We continue the study of these graph classes by considering coloring, clique, and isomorphism problems on $H$-graphs. We show that for any fixed $H$ containing a certain 3-node, 6-edge multigraph as a minor that the clique problem is APX-hard on $H$-graphs and the isomorphism problem is isomorphism-complete. We also provide positive results on $H$-graphs. Namely, when $H$ is a cactus the clique problem can be solved in polynomial time. Also, when a graph $G$ has a Helly $H$-representation, the clique problem can be solved in polynomial time. Finally, we observe that one can use treewidth techniques to show that both the $k$-clique and list $k$-coloring problems are FPT on $H$-graphs. These FPT results apply more generally to treewidth-bounded graph classes where treewidth is bounded by a function of the clique number." @default.
- W4300451657 created "2022-10-03" @default.
- W4300451657 creator A5013404440 @default.
- W4300451657 creator A5018484255 @default.
- W4300451657 date "2017-06-02" @default.
- W4300451657 modified "2023-09-30" @default.
- W4300451657 title "Combinatorial Problems on $H$-graphs" @default.
- W4300451657 doi "https://doi.org/10.48550/arxiv.1706.00575" @default.
- W4300451657 hasPublicationYear "2017" @default.
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