How to reduce the hamiltonian path problem to 1/2 hamiltonian path problem
Double the graph size: make two clones of the input $G_1,G_2$ and now create the (not connected) graph $\hat G$ that will consist of the two clones $G_1,G_2$. Now a half-hamiltonian path in $\hat G$ is either going through all $G_1$ or all $G_2$ (but not both, since they are not connected) and thus would be a hamiltonian path in $G$.