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
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.