Sphere Representations, Stacked Polytopes, and the Colin de Verdière Number of a Graph

Lon Mitchell, Lynne Yengulalp


We prove that a $k$-tree can be viewed as a subgraph of a special type of $(k+1)$-tree that corresponds to a stacked polytope and that these "stacked'' $(k+1)$-trees admit representations by orthogonal spheres in $\mathbb{R}^{k+1}$. As a result, we derive lower bounds for Colin de Verdière's $\mu$ of complements of partial $k$-trees and prove that $\mu(G) + \mu(\overline{G}) \geq |G| - 2$ for all chordal $G$.


Colin de Verdière invariant; Chordal graphs; Sphere representations; Stacked polytopes

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