Intrinsically projectively linked graphs

TL;DR

This paper investigates intrinsically projectively linked graphs, identifying new minimal examples, establishing conditions for their existence, and analyzing embeddings in projective space to understand linking properties.

Contribution

It determines the nonexistence of minor-minimal IPL graphs on 16 edges, finds new minimal IPL graphs via $ riangle$-Y exchanges, and characterizes conditions for IPL in projective planar graphs.

Findings

No minor-minimal IPL graphs on 16 edges exist.

New minor-minimal IPL graphs identified through $ riangle$-Y exchanges.

Conditions for $f(G + ar{K_{2}})$ to have no nonsplit link established.

Abstract

A graph is intrinsically projectively linked (IPL) if its every embedding in projective space contains a nonsplit link. Some minor-minimal IPL graphs have been found previously. We determine that no minor-minimal IPL graphs on 16 edges exists and identify new minor-minimal IPL graphs by applying $\Delta-Y$ exchanges to $K_{7}-2e$. We prove that for a nonouter-projective-planar graph $G$, $G+\bar{K}_{2}$ is IPL and describe the necessary and sufficient conditions on a projective planar graph $G$ such that $G+\bar{K}_{2}$ is IPL. Lastly, we deduce conditions for $f(G + \bar{K_{2}})$ to have no nonsplit link, where $G$ is projective planar, $\bar{K_{2}} = \{w_{0},w_{1}\}$, and $f(G + \bar{K_{2}})$ is the embedding onto $\mathbb{R}P^{3}$ with $f(G)$ in $z=0$, $w_{0}$ above $z=0$, and $w_{1}$ below $z=0$ such that every edge connecting ${w_{0},w_{1}}$ to $G$ avoids the boundary of the 3-ball, whose antipodal points are identified to obtain projective space.