Comments on "$\mathcal{O}(m\cdot n)$ algorithms for the recognition and isomorphism problems on circular-arc graphs"

TL;DR

This paper critically examines previous claims of $ ext{O}(m imes n)$ algorithms for recognizing and testing isomorphism of circular-arc graphs, revealing flaws in all three proposed results including decomposition trees and recognition algorithms.

Contribution

The paper provides a critical analysis showing that the earlier claimed $ ext{O}(m imes n)$ algorithms and decomposition structures for circular-arc graphs are incorrect.

Findings

Hsu's isomorphism algorithm is incorrect.

Decomposition trees for circular-arc graphs are flawed.

Recognition algorithms for circular-arc graphs are flawed.

Abstract

In the work [$\mathcal{O}(m\cdot n)$ algorithms for the recognition and isomorphism problems on circular-arc graphs, SIAM J. Comput. 24(3), 411--439, (1995)], Wen-Lian Hsu claims three results concerning the class of circular-arc graphs: - the design of so-called \emph{decomposition trees} that represent the structure of all normalized intersection models of circular-arc graphs, - an $\mathcal{O}(m\cdot n)$ recognition algorithm for circular-arc graphs, - an $\mathcal{O}(m\cdot n)$ isomorphism algorithm for circular-arc graphs. In [Discrete Math. Theor. Comput. Sci., 15(1), 157--182, 2013] Curtis, Lin, McConnell, Nussbaum, Soulignac, Spinrad, and Szwarcfiter showed that Hsu's isomorphism algorithm is incorrect. In this note, we show that the other two results -- namely, the construction of decomposition trees and the recognition algorithm -- are also flawed.