Constructing Uniquely Realizable Graphs

Abstract

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 $$\mathbb R ^d$$ , 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 which can be solved 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 describe a general framework to construct uniquely k-colorable graphs. These graphs have the extra property of being uniquely vector k-colorable. We give a deterministic explicit construction of such a family using Cayley expander graphs.