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

TL;DR

It is shown that Hsu's isomorphism algorithm is incorrect, and the construction of decomposition trees and the recognition algorithm -- namely, the construction of decomposition trees and the recognition algorithm -- are also 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.

Figures (4)

• Figure 2.1: From left to right: $u_1$ and $u_2$ are independent, $u_1$ contains $u_2$, $u_1$ is contained in $u_2$, $u_1$ and $u_2$ cover the circle, and $u_1$ and $u_2$ strictly overlap.
• Figure 2.2: Counterexample \ref{['count:main_counter']}: $V(G_c)$ is a neighbourhood module, $M_1,M_2,M_3,M_4$ are all parallel children of $V(G_c)$ in the modular decomposition tree of $G_c$, and $M_1$ and $M_4$ are not consistent.
• Figure 3.1: Counterexample to Claim A.
• Figure 3.2: Counterexample to Claim B.