How to find two disjoint spanning trees of an undirected graph

Question

Is there any applicable approach to find two disjoint spanning trees of an undirected graph or to check if a certain graph has two disjoint spanning trees

Answer 1

Assuming that the desire is to find spanning trees with disjoint edge sets, what about:

1. Given a graph G determining the minimum spanning tree A of G.
2. Defining B = G - A by deleting all edges from G that also lie in A.
3. Checking if B is connected.

The nature of a minimum spanning tree somehow makes me intuitively believe that choosing it as one of the two spanning trees gives you maximum freedom in constructing the other (that hopefully turns out to be edge disjunctive).

What do You guys think?

edit

The above algorithm makes no sense as a spanning tree is a tree and therefore needs to be acyclic. But there is no guarantee that B = G - A is acyclic.

However, this observations (thx@Tormer) led me to another idea:

1. Given a graph G determine the minimum spanning tree A of G.
2. Define B = (V[G], E[G] \ E[A]) where V[G] describes the vertices of G and E[G] describes the edges of G (A respectively).
3. Determine, if B has a spanning tree.

It could very well be that the above algorithm fails although G indeed has two edge disjunctive spanning trees - just no one of them is G's minimum spanning tree. I can't judge this (now), so I'm asking for Your opinion if it's wise to always chose the minimum spanning tree as one of the two.

Answer 2

Not sure it helps much in the applicable side but Tutte [1961a] and Nash-Williams [1961] independently characterized graphs having k pairwise edge-disjoint spanning trees:

A graph G has k pairwise edge-disjoint spanning trees iff for every partition of the vertices of G into r sets, there are at least k(r-1) edges of G whose endpoints are in different sets of the partition.

Use k=2 and it may give you a lead for your needs.

Answer 3

This is an example of Matroid union. Consider the graphic matroid where the basis are given by the spanning trees. Now the union of this matroid with itself is again a matroid. Your question is about the size of the basis of this matroid. (whether there exist a basis of size $2(|V|-1)$.

The canonical algorithm for this is Matroid partitioning algorithm. There exist an algorithm which does does the following: It maintains a set of edges with a partitioning into two forests. At each step given a new edge $e$, it decides whether there exist a reshuffling of the current partition into a new partition such that the new edge can be added to the set and the partition remains independent. And if not, it somehow will provide a certificate that it cannot.

For details look at a course in Comb. Optimization or the book by Schriver.

Answer 4

According to A Note on Finding Minimum-Cost Edge-Disjoint Spanning Trees, this can be solved in O(k²n²) where k is the number of disjoint spanning trees, and n is the number of vertices.

Unfortunately, all but the first page of the article is behind a paywall.