Links in projective planar graphs

TL;DR

This paper studies special classes of graphs embedded in the projective plane, characterizing minimal graphs that necessarily contain certain linked cycles, advancing understanding of topological graph properties.

Contribution

It provides partial characterizations of minor-minimal separating and IPPI3L graphs, introducing new classifications and necessary conditions for these graph types.

Findings

Characterized minor-minimal separating projective planar graphs.

Classified all minor-minimal IPPI3L graphs with three or more components.

Identified many minor-minimal IPPI3L graphs with fewer components.

Abstract

A graph $G$ is nonseparating projective planar if $G$ has a projective planar embedding without a nonsplit link. Nonseparating projective planar graphs are closed under taking minors and are a superclass of projective outerplanar graphs. We partially characterize the minor-minimal separating projective planar graphs by proving that given a minor-minimal nonouter-projective-planar graph $G$, either $G$ is minor-minimal separating projective planar or $G \dot\cup K_{1}$ is minor-minimal weakly separating projective planar, a necessary condition for $G$ to be separating projective planar. One way to generalize separating projective planar graphs is to consider type I 3-links consisting of two cycles and a pair of vertices. A graph is intrinsically projective planar type I 3-linked (IPPI3L) if its every projective planar embedding contains a nonsplit type I 3-link. We partially characterize minor-minimal IPPI3L graphs by classifying all minor-minimal IPPI3L graphs with three or more components, and finding many others with fewer components.