Graph Isomorphism Parameterized by Elimination Distance to Bounded Degree

TL;DR

This paper introduces a new parameter called elimination distance to bounded degree for graphs, showing that graph isomorphism can be efficiently solved when parameterized by this measure, extending previous fixed-parameter tractability results.

Contribution

It generalizes deletion distance to triviality by defining elimination distance to bounded degree, proving that graph isomorphism is fixed-parameter tractable with respect to this new parameter.

Findings

Graph canonisation is FPT when parameterized by elimination distance to bounded degree.

Extends fixed-parameter tractability results for graph isomorphism.

Provides a new framework for analyzing graph isomorphism complexity.

Abstract

A commonly studied means of parameterizing graph problems is the deletion distance from triviality (Guo et al. 2004), which counts vertices that need to be deleted from a graph to place it in some class for which efficient algorithms are known. In the context of graph isomorphism, we define triviality to mean a graph with maximum degree bounded by a constant, as such graph classes admit polynomial-time isomorphism tests. We generalise deletion distance to a measure we call elimination distance to triviality, based on elimination trees or tree-depth decompositions. We establish that graph canonisation, and thus graph isomorphism, is FPT when parameterized by elimination distance to bounded degree, extending results of Bouland et al. (2012).