is it possible to find the maximal min cut in polynomial time?

Question

A maximal minimum cut is a minimum capacity cut with the largest number of edges.

Answer 1

This problem is NP-hard if 0 weight is allowed. We can reduce Not-All-Equal 3SAT to the decision version of this problem.

Given an instance of Not-All-Equal 3SAT with $n$ variables and $m$ clauses, for each variable $x_i$, we create two vertices $v_i$ and $v_i'$ with an edge between them for each variable. In addition, for each clause, for example, $x_1\vee x_2\vee \neg x_3$, we add three edges among $v_1,v_2,v_3'$ (thus they form a triangle). All edges have weight 0. Now we can see there is a minimum cut (in fact, every cut is a minimum cut) with at least $n+2m$ edges if and only if the Not-All-Equal 3SAT instance is satisfiable.