Finding a Spanning Tree Using other Spanning Trees of =(,)

Question

I am having trouble coming up with a polynomial time algorithm to solve the following problem:

Let =(,) be an undirected and unweighted graph with vertices. Let _1,_2,...,_ be =−1 distinct spanning trees of . Find a polynomial time algorithm that finds a spanning tree
=(,_) in that contains at least one edge from each spanning tree _.

I would really appreciate any help on this!

Answer 1

Here is a basic solution.

//if the graph is given in input:
//    If any bridge exists in the graph:
//        return any spanning tree from the list of input spanning trees
    
while the size of result set is less than |V|-1:
    for every tree in the list of spanning trees:
        for every edge in the tree:
            if edge is not a part of result set:
                if adding this edge to the result set doesn't form a cycle:
                    add the edge to result set
                    break
        if size of result set is V-1:
            break

• Size of list of spanning trees < |V|-1: while loop is required
• Size of list of spanning trees > |V|-1: a proof would be required that "the result set returned by above algorithm includes at least one edge from all the remaining trees in the list of spanning trees". I'll try this proof at leisure.

If for the given graph |E| < 2*(|V|-1) or the size list of spanning trees < |V|, then the above algorithm definitely works.

An interesting case is when |E| >= 2*(|V|-1). As shown in the image below, for some input list of spanning trees L, say the set of green edges is the result set returned by the above algorithm. A spanning tree of Blue edges could be formed without using any of the green edges.

Just posted this answer so that it works as some basic ground work for people trying to answer this question. Otherwise, could be treated as brute force solution.

Answer 2

Spanning trees have matroid structure. Therefore the following greedy algorithm works: starting with an empty forest, for each input spanning tree, extend the forest by any one tree edge that does not create a cycle. Correctness follows more or less directly from the augmentation property and the definition of independence.