$st$-path with fewest leaving edges Given an undirected unweighted  multigraph $G=(V,E)$ and $s,t \in V$ find a simple $st$-path $P$ s.t. the number of edges leaving $P$ (i.e. the edges with exaclty one endpoint in $P$ ) is minimized.

Does anybody have any idea how to solve this?
I first thought about replacing edges by two arcs and weight them according to some node degrees but the edges staying in $P$ make life difficult.
Suppose input graph is simple (without parallel edges), now, If you solve your problem, you can check existence of Hamiltonian path, actually if s-t path of cost zero for some pair exists means there is Hamiltonian path and you can check this for all pair of vertices as (s,t), so you are unlucky to have polynomial time algorithm except $P=NP$.