Links in projective planar graphs

TL;DR

This work extends the theory of nonseparating graphs from the plane to the projective plane by classifying minor-minimal separating projective-planar graphs and analyzing their connections to homogeneous cycle structures in $\mathbb{R}P^2$. It introduces and partially characterizes intrinsically projective-planar type I 3-linked graphs (IPPI3L), identifying several minimal three-component and two-component constructions and exploring gluing operations and planarity constraints. The authors leverage cycle-homology results, minor-closed properties, and the finite set of minor-minimal nonouter-projective-planar graphs to distinguish between separating, weakly separating, and strongly nonseparating embeddings, providing a framework that links topological linking in projective space with graph-minor theory. The findings lay groundwork for a fuller classification of IPPI3L graphs and related minor-minimal structures, with open questions about the complete IPPI3L set and the nature of IPPII3L graphs.

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.

Figures (45)

• Figure 1: Adding a H-path $H_{1}$ to the cycle
• Figure 2: Existence of a path connecting $p_{i}$, $p_{i+1}$ intersecting the boundary only at $p_{i}$, $p_{i+1}$
• Figure 3: Reduced number of crossing with the boundary
• Figure 4: Reduced number of crossings of $C_{2}$ with the boundary by applying isotopy to $L$
• Figure 5: $\beta_1$

Theorems & Definitions (97)

• Lemma 3.1
• Theorem 3.1
• proof
• Lemma 3.2
• proof
• Theorem 3.2
• proof
• Corollary 3.1
• Lemma 3.3: Glover et al Glover
• Theorem 3.3