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We show that if $G=(V,E)$ is a 4-connected flat graph, then any realsymmetric $V\times V$ matrix $M$ with exactly one negative eigenvalue andsatisfying, 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 StrongArnold 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 Euclideanspace such that the image of any circuit is the boundary of some disk disjointfrom the image of the remainder of the graph.)  This applies to the Colin de Verdi\`ere graph parameter, and extends similarresults for 2-connected outerplanar graphs and 3-connected planar graphs.

The Strong Arnold Property for 4-connected flat graphs