# What is the Advantage of 1-trees over Vertex Splitting?

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?