Constructions stemming from non-separating planar graphs and their Colin de Verdière invariant

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 Verdière invariant of the complement $cG$ satisfies $μ(cG) \le n-4$. We show this to be an equality. As a consequence, the conjecture of Kotlov, Lovàsz, and Vempala that for a simple graph $G$, $μ(G)+μ(cG)\ge n-2$ 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.

Figures (12)

• Figure 1: $\nabla Y-$ and $Y\nabla-$moves
• Figure 2: (a) elongated prism with only two edges subdivided; (b) planar graph obtained by deleting the vertices $t$ and $w$ of $H+P_2$
• Figure 3: (a) the graph $P'$ obtained by subdividing once each non-triangular edge of the prism graph; (b) the graph $D_4$
• Figure 4: (a) maximal outerplanar graph with 7 vertices; (b) graph $G$, a wheel with $n$ vertices; (c) $cG\setminus \{v_7, v_8, \ldots, v_{n-1}\}$
• Figure 5: (a) elongated prism (b) subgraph induced by $\{v_1, v_3, v_5, a, b, c\}$ in $cG$

Theorems & Definitions (31)

• Theorem 1
• Theorem 2
• Theorem 3
• Theorem 4
• proof
• Theorem 5
• proof
• Theorem 6
• proof
• Theorem 7