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

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.