Graph parallel rigidity

Combinatorics Seminar
Friday, March 6, 2009 - 15:00
1 hour (actually 50 minutes)
Skiles 255
This is joint work with Alex Postnikov. Imagine that you are to build a system of space stations (graph vertices), which communicate via laser beams (edges). The edge directions were already chosen, but you must place the stations so that none of the beams miss their targets. In this talk, we let the edge directions be generic and independent, a choice that constrains vertex placement the most. For K_{3} in \mathbb{R}^{2}, the edges specify a unique triangle, but its size is arbitrary --- D_{2}(K_{3})=1 degree of freedom; we say that K_{3} is rigid in \mathbb{R}^{2}. We call D_{n}(G) the degree of parallel rigidity of the graph for generic edge directions. We found an elegant combinatorial characterization of D_{n}(G) --- it is equal to the minimal number of edges in the intersection of n spanning trees of G. In this talk, I will give a linear-algebraic proof of this result, and of some other properties of D_{n}(G). The notion of parallel graph rigidity was previously studied by Whiteley and Develin-Martin-Reiner. The papers worked with the generic parallel rigidity matroid; I will briefly compare our results in terms of D_{n}(G) with the previous work.