Canonizing Graphs of Bounded Tree Width in Logspace

TL;DR

This paper demonstrates that graphs with bounded tree width can be canonized and their isomorphism problem decided efficiently in logarithmic space, advancing understanding of graph isomorphism complexity.

Contribution

It proves that graphs of bounded tree width can be canonized in logspace, establishing their isomorphism problem as solvable within this complexity class.

Findings

Graphs of bounded tree width can be canonized in logspace.

The isomorphism problem for these graphs is decidable in logspace.

This class of graphs is among the few with a precisely determined isomorphism complexity.

Abstract

Graph canonization is the problem of computing a unique representative, a canon, from the isomorphism class of a given graph. This implies that two graphs are isomorphic exactly if their canons are equal. We show that graphs of bounded tree width can be canonized by logarithmic-space (logspace) algorithms. This implies that the isomorphism problem for graphs of bounded tree width can be decided in logspace. In the light of isomorphism for trees being hard for the complexity class logspace, this makes the ubiquitous class of graphs of bounded tree width one of the few classes of graphs for which the complexity of the isomorphism problem has been exactly determined.