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

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 Rk+1. As a result, we derive lower bounds for Colin de Verdière’s μ of complements of partial k-trees and prove that μ(G) + μ(G) ≥ |G| − 2 for all chordal G. Yves Colin de Verdière’s graph invariant μ is defined as the maximum nullity over a special class of real symmetric matrices [3]. Among its many interesting properties are that μ is minor monotone and μ(G) ≤ 3 if and only if G is planar. These properties are collected in a recent survey by László Lovász [13] based on an earlier paper with Hein van der Holst and Alexander Schrijver [22].