Circle Graph Isomorphism in Almost Linear Time

TL;DR

This paper introduces an efficient algorithm for circle graph isomorphism that operates in almost linear time, significantly improving previous methods by leveraging minimal split decomposition and advanced recognition algorithms.

Contribution

The paper presents a novel almost linear time algorithm for circle graph isomorphism, improving upon the previous quadratic time solutions using minimal split decomposition.

Findings

Algorithm runs in $O((n+m)\alpha(n+m))$ time

Improves previous $O(nm)$ algorithm

Utilizes minimal split decomposition and advanced recognition

Abstract

Circle graphs are intersection graphs of chords of a circle. In this paper, we present a new algorithm for the circle graph isomorphism problem running in time $O((n+m)\alpha(n+m))$ where $n$ is the number of vertices, $m$ is the number of edges and $\alpha$ is the inverse Ackermann function. Our algorithm is based on the minimal split decomposition [Cunnigham, 1982] and uses the state-of-art circle graph recognition algorithm [Gioan, Paul, Tedder, Corneil, 2014] in the same running time. It improves the running time $O(nm)$ of the previous algorithm [Hsu, 1995] based on a similar approach.