# Canonical Representations for Circular-Arc Graphs Using Flip Sets

## TL;DR

This paper introduces a polynomial-time method for computing canonical representations of uniform circular-arc graphs using flip sets, and explores complexity bounds for broader classes of CA graphs.

## Contribution

It presents a new polynomial-time approach for uniform CA graphs and links the canonical representation problem to restricted CA matrices, advancing graph canonization theory.

## Key findings

- Canonical representations for uniform CA graphs can be computed in polynomial time.
- The canonical representation problem for CA graphs reduces to restricted CA matrices.
- CA graphs without induced 4-cycles can be canonized in logspace.

## Abstract

We show that computing canonical representations for circular-arc (CA) graphs reduces to computing certain subsets of vertices called flip sets. For a broad class of CA graphs, which we call uniform, it suffices to compute a CA representation to find such flip sets. As a consequence canonical representations for uniform CA graphs can be obtained in polynomial-time. We then investigate what kind of CA graphs pose a challenge to this approach. This leads us to introduce the notion of restricted CA matrices and show that the canonical representation problem for CA graphs is logspace-reducible to that of restricted CA matrices. As a byproduct, we obtain the result that CA graphs without induced 4-cycles can be canonized in logspace.