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

TL;DR

This paper introduces simple, fast algorithms to determine if a connected undirected graph with 2(V-1) edges can be decomposed into two edge disjoint spanning trees, improving on previous complex methods.

Contribution

The authors present the first nearly linear time algorithms for decomposing certain undirected graphs into edge disjoint spanning trees using simple graph reduction techniques.

Findings

Algorithms are simpler and easier to implement than previous matroid-based methods.

Achieve asymptotically faster running times, approaching linear time.

Enable recognition of minimally rigid (Laman) graphs in almost linear time.

Abstract

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.