Decide whether there is a MST that contains some edges of 2 distinct edge sets

Question

Let G = (V, E) be a weighted, connected and undirected graph. Let T1 and T2 be 2 different MST's. Suppose we can write E = (A1 U B U A2) such that:
B is the intersection of the edges of T1 and T2, and
A1 = T1 - B
A2 = T2 - B

Assuming that every MST T in G contains all the edges of B, find an algorithm that decides whether there is a MST T that contains at least one edge in A1 and at least one edge in A2.

Edit: I've dropped the part that was here. I think that it does more harm than good.

Answer 1

you should sort your edge that the red edge is prefer to blue edge for choose.then you can use any MST algorithm same as Prim's algorithm :

If a graph is empty then we are done immediately. Thus, we assume otherwise. The algorithm starts with a tree consisting of a single vertex, and continuously increases its size one edge at a time, until it spans all vertices. Input: A non-empty connected weighted graph with vertices V and edges E (the weights can be negative). Initialize: Vnew = {x}, where x is an arbitrary node (starting point) from V, Enew = {} Repeat until Vnew = V: Choose an edge {u, v} with minimal weight such that u is in Vnew and v is not (if there are multiple edges with the same weight, any of them may be picked) Add v to Vnew, and {u, v} to Enew Output: Vnew and Enew describe a minimal spanning tree