Graph theory problem

Let G have even no. of nodes. I want to eventually merge all nodes into
1. At each step, I will merge any two nodes. The cost of each merge is
the weight
of the edge common to the two vertices. The cost at each step is the
sum of the cost of all such merges. I want to minimize the cost over
all the steps.
Does this sound like any common graph theoretic problem? You can assume
edges are undirected and the graph to be fully connected if that makes
life easy.
Thanks in advance

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