Circular Separation Dimension of a Subclass of Planar Graphs

TL;DR

This paper investigates the circular separation dimension for specific subclasses of planar graphs, proving that 2-outerplanar graphs have a dimension of 2 and series-parallel graphs have a dimension at most 2.

Contribution

It establishes the exact and upper bound values of the circular separation dimension for 2-outerplanar and series-parallel graphs, respectively.

Findings

2-outerplanar graphs have a circular separation dimension of 2

Series-parallel graphs have a circular separation dimension at most 2

Introduces bounds for circular separation dimension in specific planar graph subclasses

Abstract

A pair of non-adjacent edges is said to be separated in a circular ordering of vertices, if the endpoints of the two edges do not alternate in the ordering. The circular separation dimension of a graph $G$, denoted by $\pi^\circ(G)$, is the minimum number of circular orderings of the vertices of $G$ such that every pair of non-adjacent edges is separated in at least one of the circular orderings. This notion is introduced by Loeb and West in their recent paper. In this article, we consider two subclasses of planar graphs, namely $2$-outerplanar graphs and series-parallel graphs. A $2$-outerplanar graph has a planar embedding such that the subgraph obtained by removal of the vertices of the exterior face is outerplanar. We prove that if $G$ is $2$-outerplanar then $\pi^\circ(G) = 2$. We also prove that if $G$ is a series-parallel graph then $\pi^\circ(G) \leq 2$.