Towards an Optimal Algorithm for Recognizing Laman Graphs

A graph G with n vertices and m edges is a Laman graph, or equivalently a generically minimally rigid graph, if m = 2n - 3 and every induced subset of k vertices spans at most 2k - 3 edges. Laman graphs play a fundamental role in rigidity theory. We discuss the verification problem: Given a graph G with n vertices, decide if it is Laman. We present an algorithm that recognizes Laman graphs in O(T_st(n) + n log n) time, where T_st(n) is the best time to extract two edge disjoint spanning trees from a graph with n vertices and 2n - 2 edges, or decide no such trees exist. So far, it is known that T_st(n) is O(n^3/2radic(log n)).