Connectivity and matching extendability of optimal $1$-embedded graphs on the torus

TL;DR

This paper characterizes the connectivity levels and matching extendability of optimal 1-toroidal graphs, revealing specific connectivity values and their implications for graph extendability on the torus.

Contribution

It provides a complete characterization of connectivity and matching extendability for optimal 1-toroidal graphs, including the non-existence of certain connectivity levels.

Findings

O1TGs with connectivity exactly 4, 5, 6, and 8 are characterized.

No O1TG has connectivity exactly 7.

The extendability of matchings in O1TGs is fully characterized for 1-, 2-, and 3-extendability.

Abstract

In this paper, we discuss optimal $1$-toroidal graphs (abbreviated as O1TG), which are drawn on the torus so that every edge crosses another edge at most once, and has $n$ vertices and exactly $4n$ edges. We first consider connectivity of O1TGs, and give the characterization of O1TGs having connectivity exactly $k$ for each $k\in \{4, 5, 6, 8\}$. In our argument, we also show that there exists no O1TG having connectivity exactly $7$. Furthermore, using the result above, we discuss extendability of matchings, and give the characterization of $1$-, $2$- and $3$-extendable O1TGs in turn.