Towards an optimal algorithm for recognizing Laman graphs

Abstract

A graph G with n vertices and m edges is a generically minimally rigid graph (Laman 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 logn) 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^3/2√{logn}).