The matching extendability of optimal $1$-embedded graphs on the projective plane

TL;DR

This paper investigates the matching extendability properties of optimal 1-embedded graphs on the projective plane, providing characterizations of their extendability and connectivity, and identifying non-extendable configurations.

Contribution

It introduces new characterizations of 1-extendability and 2-extendability for optimal 1-projective plane graphs, and analyzes their connectivity and non-extendable edge configurations.

Findings

Even order O1PPGs are 1-extendable.

Characterization of 2-extendable O1PPGs via non-crossing separating cycles.

Identification of non-extendable independent edges in graphs with connectivity 5.

Abstract

In this paper, we discuss matching extendability of optimal $1$-projective plane graphs (abbreviated as O1PPG), which are drawn on the projective plane $P^2$ so that every edge crosses another edge at most once, and has $n$ vertices and exactly $4n- 4$ edges. We first show that every O1PPG of even order is $1$-extendable. Next, we characterize $2$-extendable O1PPG's in terms of a separating cycle consisting of only non-crossing edges. Moreover, we characterize O1PPG's having connectivity exactly $5$. Using the characterization, we further identify three independent edges in those graphs that are not extendable.