Consider a flow network $G$. Let $(S, T)$ be a min-cut of $G$. Let $(u, v)$ be an edge that crosses the cut from $S$ to $T$. Claim: increasing the capacity of $(u, v)$ causes the value of the maximum flow in $G$ to increase.
I'm 99% sure this is false, but I'm struggling to disprove it. Can anyone give me some tips to find a counter-example?
I thought of making a graph with a cut and having two edges run through this cut such that one is the minimum and one is greater; increasing the greater one won't change the max-flow by the min-cut theorem.