Testing isomorphism of chordal graphs of bounded leafage is fixed-parameter tractable

TL;DR

This paper proves that testing isomorphism of chordal graphs with bounded leafage is fixed-parameter tractable, advancing understanding of the graph isomorphism problem for specific graph classes.

Contribution

It establishes fixed-parameter tractability for chordal graph isomorphism based on leafage and introduces fixed-parameter algorithms for hypergraph automorphism problems.

Findings

Chordal graph isomorphism is fixed-parameter tractable with leafage as parameter.

Automorphism group computation for order-k hypergraphs is fixed-parameter tractable.

Provides new algorithms for isomorphism testing in hypergraphs with fixed parameters.

Abstract

The computational complexity of the graph isomorphism problem is considered to be a major open problem in theoretical computer science. It is known that testing isomorphism of chordal graphs is polynomial-time equivalent to the general graph isomorphism problem. Every chordal graph can be represented as the intersection graph of some subtrees of a representing tree, and the leafage of a chordal graph is defined to be the minimum number of leaves in a representing tree for it. We prove that chordal graph isomorphism is fixed parameter tractable with leafage as parameter. In the process we introduce the problem of isomorphism testing for higher-order hypergraphs and show that finding the automorphism group of order-$k$ hypergraphs with vertex color classes of size $b$ is fixed parameter tractable for any constant $k$ and $b$ as fixed parameter.