# Efficient Isomorphism for $S_d$-graphs and $T$-graphs

## TL;DR

This paper develops fixed-parameter tractable algorithms for the graph isomorphism problem in specific classes of intersection graphs derived from fixed graphs, notably $S_d$-graphs and $T$-graphs, addressing open problems and extending previous methods.

## Contribution

It introduces FPT algorithms for isomorphism testing of $S_d$-graphs and $T$-graphs, and extends these to proper graphs, solving open problems and providing new computational techniques.

## Key findings

- FPT algorithm for $S_d$-graph isomorphism based on group machinery
- XP algorithm for $T$-graph isomorphism parameterized by tree size
- Efficient algorithms for proper $S_d$- and $T$-graphs

## Abstract

An $H$-graph is one representable as the intersection graph of connected subgraphs of a suitable subdivision of a fixed graph $H$, introduced by Bir\'{o}, Hujter and Tuza (1992). An $H$-graph is proper if the representing subgraphs of $H$ can be chosen incomparable by the inclusion. In this paper, we focus on the isomorphism problem for $S_d$-graphs and $T$-graphs, where $S_d$ is the star with $d$ rays and $T$ is an arbitrary fixed tree. Answering an open problem of Chaplick, T\"{o}pfer, Voborn\'{\i}k and Zeman (2016), we provide an FPT-time algorithm for testing isomorphism and computing the automorphism group of $S_d$-graphs when parameterized by~$d$, which involves the classical group-computing machinery by Furst, Hopcroft, and Luks (1980). We also show that the isomorphism problem of $S_d$-graphs is at least as hard as the isomorphism problem of posets of bounded width, for which no efficient combinatorial-only algorithm is known to date. Then we extend our approach to an XP-time algorithm for isomorphism of $T$-graphs when parameterized by the size of $T$. Lastly, we contribute a simple FPT-time combinatorial algorithm for isomorphism testing in the special case of proper $S_d$- and $T$-graphs.