Solving the Canonical Representation and Star System Problems for Proper Circular-Arc Graphs in Log-Space

TL;DR

This paper introduces a logspace algorithm for constructing canonical models of proper circular-arc graphs and related classes, enabling efficient recognition and isomorphism testing in low computational complexity.

Contribution

It provides the first logspace algorithms for canonical intersection models and the Star System Problem for these graph classes, advancing graph isomorphism research.

Findings

Canonical models of proper circular-arc graphs can be constructed in logspace.

Recognition and isomorphism problems for these graphs are solvable in logspace.

The Star System Problem can be solved in logspace for multiple graph classes.

Abstract

We present a logspace algorithm that constructs a canonical intersection model for a given proper circular-arc graph, where `canonical' means that models of isomorphic graphs are equal. This implies that the recognition and the isomorphism problems for this class of graphs are solvable in logspace. For a broader class of concave-round graphs, that still possess (not necessarily proper) circular-arc models, we show that those can also be constructed canonically in logspace. As a building block for these results, we show how to compute canonical models of circular-arc hypergraphs in logspace, which are also known as matrices with the circular-ones property. Finally, we consider the search version of the Star System Problem that consists in reconstructing a graph from its closed neighborhood hypergraph. We solve it in logspace for the classes of proper circular-arc, concave-round, and co-convex graphs.