Towards an optimal algorithm for recognizing Laman graphs

Laman graphs are fundamental to rigidity theory. A graph G with n vertices and m edges is a generic minimally rigid graph (Laman graph), if m=2n-3 and every induced subset of k vertices spans at most 2k-3 edges. We consider the verification problem: Given a graph G with n vertices, decide if it is Laman. We present an algorithm that takes O(T(n)+n log n) time, where T(n) is the best time to extract two edge disjoint spanning trees from G or decide no such trees exist. Our algorithm exploits a known construction called red-black hierarchy (RBH), that is a certificate for Laman graphs. First, we show how to verify if G admits an RBH and argue this is enough to conclude whether G is Laman or not. Second, we show how to construct the RBH using a two steps procedure that is simple and easy to implement. Finally, we point out some difficulties in using red-black hierarchies to compute a Henneberg construction, which seem to imply super-quadratic time algorithms when used for embedding a planar Laman graph as a pointed pseudo-triangulation.