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 R k + 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 .
The Electronic Journal of Combinatorics
Mitchell, L. & Yengulalp, L. (2016). Sphere representations, stacked polytopes, and the Colin de Verdière number of a graph. The Electronic Journal of Combinatorics, 23(1), P1.9. https://doi.org/10.37236/4444.