Let $G$ be a weighted digraph, $s$ and $t$ be two vertices of $G$, and $t$ is reachable from $s$. The logical $s$-$t$ min-cut (LSTMC) problem states how $t$ can be made unreachable from $s$ by removal of some edges of $G$ where (a) the sum of weights of the removed edges is minimum and (b) all outgoing edges of any vertex of $G$ cannot be removed together. If we ignore the second constraint, called the logical removal, the LSTMC problem is transformed to the classic $s$-$t$ min-cut problem. The logical removal constraint applies in situations where non-logical removal is either infeasible or undesired. Although the $s$-$t$ min-cut problem is solvable in polynomial time by the max-flow min-cut theorem, this paper shows the LSTMC problem is NP-Hard, even if $G$ is a DAG with an out-degree of two. Moreover, this paper shows that the LSTMC problem cannot be approximated within $alpha log n$ in a DAG with $n$ vertices for some constant $alpha$. The application of the LSTMC problem is also presented intest case generation of a computer program.