Question

Let an edge (p,q) that has the smallest (minimum) weight in a connected graph G=(V,E,w) where w is the weight function. Show that the edge (p,q) belongs to some minimum spanning tree of G.

Answer 1

Here the question given is for the edge (p,q) which has minimum weight in the connected graph. We have to show that the edge (p,q) belongs to some minimum spanning tree of G.

Let us assume that the given minimum edge be e and that the given statement is not true. Then our assuming statement will be the edge,e must not present in any of the minimum spanning tree of G. Create the minimum spanning tree assuming that there is no edge,e in the original given graph, G. Then we get tree(s) with sum of weights is minimum. Then in the resultant tree, remove the minimum edge present and add the given minimum edge,e. Which in turn gives us the better minimum spanning tree than the previous. So it shows that we assumed the contradiction about edge,e must not be there in any MST of G.

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