Edge Disjoint Spanning Trees in an Undirected Graph with E=2(V-1).

Given a connected undirected graph G = [V; E] where |E| =2(|V| -1), we present two algorithms to check if G can be decomposed into two edge disjoint spanning trees, and provide such a decomposition when it exists. Unlike previous algorithms for finding edge disjoint spanning trees in general undirected graphs, based on matroids and complex in description, our algorithms are based on simple graph reduction techniques and thus easy to describe and implement. Moreover, the running time for our solutions is asymptotically faster. Specifically, ours are the first algorithms to achieve a running time that is a polylog factor from linear, approaching the 1974 linear time algorithm of Robert E. Tarjan for directed graphs. A direct implication of our result is that minimally rigid graphs, also called Laman graphs, can be recognized in almost linear time, thus answering a long standing open problem.