Recognition and Isomorphism of Proper $\boldsymbol{U}$-graphs in FPT-time

TL;DR

This paper studies recognition and isomorphism problems for proper U-graphs, providing fixed-parameter tractable algorithms when U is unicyclic, and establishing complexity results for broader classes.

Contribution

It introduces FPT algorithms for recognizing and testing isomorphism of proper U-graphs when U is unicyclic, and proves NP-hardness and complexity bounds for general cases.

Findings

Recognition of proper U-graphs is NP-hard.

FPT algorithms are developed for recognition and isomorphism when U is unicyclic.

Isomorphism for non-unicyclic H-graphs is as hard as the general graph isomorphism problem.

Abstract

An $H$-graph is an intersection graph of connected subgraphs of a suitable subdivision of a fixed graph $H$. Many important classes of graphs, including interval graphs, circular-arc graphs, and chordal graphs, can be expressed as $H$-graphs, and, in particular, every graph is an $H$-graph for a suitable graph $H$. An $H$-graph is called proper if it has a representation where no subgraph properly contains another. We consider the recognition and isomorphism problems for proper $U$-graphs where $U$ is a unicylic graph. We prove that testing whether a graph is a (proper) $U$-graph, for some $U$, is NP-hard. On the positive side, we give an FPT-time recognition algorithm, parametrized by $\vert U \vert$. As an application, we obtain an FPT-time isomorphism algorithm for proper $U$-graphs, parametrized by $\vert U \vert$. To complement this, we prove that the isomorphism problem for (proper) $H$-graphs, is as hard as the general isomorphism problem for every fixed $H$ which is not unicyclic.