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It is a wellknown fact that the problem of finding an optimal Hamilton tour is equivalent to finding an optimal Hamilton path after a small modification of the problem instance, namely splitting one of the vertices $v$ into $v'$ and $v"$ and duplicating the edgeweights but setting the weight of $(v',v")$ to a sufficiently high value.

It is also easy to guarantee, that in the modified graph $v'$ and $v"$ always are leaf nodes, simply by adding a sufficiently high vertex weight to both of them, making the result equivalent to a 1-tree in the original graph.

The problem would then to determine in the Hamilton path setting the weights the other vertices such that $v'$ and $v"$ are the only leaf nodes of the spanning tree.


Question:

what is the advantage of using the 1-tree formulation in the Hamilton cycle setting over the tree formulation in the Hamilton path modification as described above?

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