# Combinatorial Problems on $H$-graphs

## TL;DR

This paper investigates the computational complexity of coloring, clique, and isomorphism problems on $H$-graphs, revealing both hardness results and polynomial-time algorithms for specific cases, and applying treewidth techniques for fixed-parameter tractability.

## Contribution

It establishes hardness results for certain $H$-graphs and provides polynomial-time algorithms for clique problems when $H$ is a cactus or has a Helly $H$-representation, along with FPT results using treewidth techniques.

## Key findings

- Clique problem is APX-hard for certain $H$-graphs.
- Clique problem is polynomial-time solvable when $H$ is a cactus.
- Both $k$-clique and list $k$-coloring are FPT on $H$-graphs.

## 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.