The Strong Arnold Property for 4-connected flat graphs

TL;DR

This paper proves that 4-connected flat graphs satisfy the Strong Arnold Property for certain matrices, extending known results for planar and outerplanar graphs, with implications for the Colin de Verdière graph parameter.

Contribution

It establishes the Strong Arnold Property for 4-connected flat graphs, broadening the class of graphs known to satisfy this property beyond planar cases.

Findings

Validates the Strong Arnold Property for 4-connected flat graphs.

Extends previous results from 2-connected outerplanar and 3-connected planar graphs.

Implications for the Colin de Verdière graph parameter.

Abstract

We show that if $G=(V,E)$ is a 4-connected flat graph, then any real symmetric $V\times V$ matrix $M$ with exactly one negative eigenvalue and satisfying, for any two distinct vertices $i$ and $j$, $M_{ij}<0$ if $i$ and $j$ are adjacent, and $M_{ij}=0$ if $i$ and $j$ are nonadjacent, has the Strong Arnold Property: there is no nonzero real symmetric $V\times V$ matrix $X$ with $MX=0$ and $X_{ij}=0$ whenever $i$ and $j$ are equal or adjacent. (A graph $G$ is {\em flat} if it can be embedded injectively in $3$-dimensional Euclidean space such that the image of any circuit is the boundary of some disk disjoint from the image of the remainder of the graph.) This applies to the Colin de Verdi\`ere graph parameter, and extends similar results for 2-connected outerplanar graphs and 3-connected planar graphs.