Deciding Circular-Arc Graph Isomorphism in Parameterized Logspace

TL;DR

This paper presents a logspace algorithm for computing canonical representations of circular-arc graphs, solving the isomorphism problem by splitting the class into uniform and non-uniform cases and handling each with tailored methods.

Contribution

It generalizes previous logspace canonization results from Helly CA graphs to all circular-arc graphs by introducing a parameterized approach for non-uniform cases.

Findings

Logspace algorithm for uniform CA graphs.

O(k + log n) space algorithm for all CA graphs.

Extension of Helly CA graph canonization to general CA graphs.

Abstract

We compute a canonical circular-arc representation for a given circular-arc (CA) graph which implies solving the isomorphism and recognition problem for this class. To accomplish this we split the class of CA graphs into uniform and non-uniform ones and employ a generalized version of the argument given by K\"obler et al (2013) that has been used to show that the subclass of Helly CA graphs can be canonized in logspace. For uniform CA graphs our approach works in logspace and in addition to that Helly CA graphs are a strict subset of uniform CA graphs. Thus our result is a generalization of the canonization result for Helly CA graphs. In the non-uniform case a specific set of ambiguous vertices arises. By choosing the parameter to be the cardinality of this set the obstacle can be solved by brute force. This leads to an O(k + log n) space algorithm to compute a canonical representation for non-uniform and therefore all CA graphs.