Isomorphism Testing for T-graphs in FPT

TL;DR

This paper proves that the isomorphism problem for T-graphs, a special class of chordal graphs, is fixed-parameter tractable when T is fixed, using decomposition and group algorithms.

Contribution

It establishes FPT complexity for T-graph isomorphism with T fixed, extending understanding of graph isomorphism in chordal graph subclasses.

Findings

T-graph isomorphism is in FPT when T is fixed

The approach combines decomposition with Babai's group algorithm

Recognition of T-graphs is not necessarily in FPT

Abstract

A T-graph (a special case of a chordal graph) is the intersection graph of connected subtrees of a suitable subdivision of a fixed tree T . We deal with the isomorphism problem for T-graphs which is GI-complete in general - when T is a part of the input and even a star. We prove that the T-graph isomorphism problem is in FPT when T is the fixed parameter of the problem. This can equivalently be stated that isomorphism is in FPT for chordal graphs of (so-called) bounded leafage. While the recognition problem for T-graphs is not known to be in FPT wrt. T, we do not need a T-representation to be given (a promise is enough). To obtain the result, we combine a suitable isomorphism-invariant decomposition of T-graphs with the classical tower-of-groups algorithm of Babai, and reuse some of the ideas of our isomorphism algorithm for S_d-graphs [MFCS 2020].