On powers of circular arc graphs

TL;DR

This paper investigates the closure properties of circular arc graphs under graph powers, establishing whether these classes are also strongly closed, and provides definitive answers to these open questions.

Contribution

The paper proves that circular arc graphs and proper circular arc graphs are strongly closed under powers, resolving previously open problems.

Findings

Circular arc graphs are strongly closed under powers.

Proper circular arc graphs are strongly closed under powers.

The open question about these classes' closure properties is settled.

Abstract

A class of graphs $\mathcal{C}$ is closed under powers if for every graph $G\in\mathcal{C}$ and every $k\in\mathbb{N}$, $G^k\in\mathcal{C}$. Also $\mathcal{C}$ is strongly closed under powers if for every $k\in\mathbb{N}$, if $G^k\in\mathcal{C}$, then $G^{k+1}\in\mathcal{C}$. It is known that circular arc graphs and proper circular arc graphs are closed under powers. But it is open whether these classes of graphs are also strongly closed under powers. In this paper we have settled these problems.