Constructions stemming from non-separating planar graphs and their Colin de Verdi\`ere invariant

TL;DR

This paper explores the properties of non-separating planar graphs, their complements, and related topological invariants, establishing new bounds on the Colin de Verdière invariant and implications for intrinsic linking and knotting.

Contribution

It introduces new constructions of linkless and knotless graphs from non-separating planar graphs and proves the Colin de Verdière invariant bounds for their complements, confirming a conjecture for specific graph classes.

Findings

Complement of maximal non-separating planar graphs is (n-7)-apex.

Colin de Verdière invariant of these complements equals n-4.

Complements of such graphs are intrinsically linked or knotted for large n.

Abstract

A planar graph $G$ is said to be non-separating if there exists an embedding of $G$ in $\mathbb{R}^2$ such that for any cycle $\mathcal{C}\subset G$, all vertices of $G\setminus \mathcal{C}$ are within the same connected component of $\mathbb{R}^2\setminus \mathcal{C}$. Dehkordi and Farr classified the non-separating planar graphs as either outerplanar graphs, subgraphs of wheel graphs, or subgraphs of elongated triangular prisms. We use maximal non-separating planar graphs to construct examples of maximal linkless graphs and maximal knotless graphs. We show that for a maximal non-separating planar graph $G$ with $n\ge 7$ vertices, the complement $cG$ is $(n-7)-$apex. This implies that the Colin de Verdi\`ere invariant of the complement $cG$ satisfies $\mu(cG) \le n-4$. We show this to be an equality. As a consequence, the conjecture of Kotlov, Lov\`asz, and Vempala that for a simple graph $G$, $\mu(G)+\mu(cG)\ge n-2$ is true for 2-apex graphs $G$ for which $G-\{u,v\}$ is planar non-separating. It also follows that complements of non-separating planar graphs of order at least nine are intrinsically linked. We prove that the complements of non-separating planar graphs $G$ of order at least ten are intrinsically knotted.