Constructing Uniquely Realizable Graphs

In the Graph Realization Problem (GRP), one is given a graph G, a set of non-negative edge-weights, and an integer d. The goal is to find, if possible, a realization of G in the Euclidian space R, such that the distance between any two vertices is the assigned edge weight. The problem has many applications in mathematics and computer science, but is NP-hard when the dimension d is fixed. Characterizing tractable instances of GRP is a classical problem, first studied by Menger in 1931. We construct two new infinite families of GRP instances whose solution can be approximated up to an arbitrary precision in polynomial time. Both constructions are based on the blow-up of fixed small graphs with large expanders. Our main tool is the Connelly’s condition in Rigidity Theory, combined with an explicit construction and algebraic calculations of the rigidity (stress) matrix. As an application of our results, we give a deterministic construction of uniquely k-colorable vertex-transitive expanders. ∗Department of Mathematics, UCLA, Los Angeles, CA 90095. Email: {pak,vilenchik}@math.ucla.edu