Complexity of the decision version of determining a min-cut
The described problem is problem A.4.4 of the book  "Limits to Parallel Computation: P-Completeness Theory" and is P-complete. It is proven via a reduction from Alternating Monotone Fanin 2, Fanout 2 CVP (problem A.1.4 of the same book).

The reduction described in this book originates from the 1990 Theoretical Computer Science paper "The Binary Network Flow Problem is Logspace Complete for P" (T. Lengauer, K. Wagner).

The reduction gets an Alternating Monotone Fanin 2, Fanout 2 CVP instance (a circuit $C$) and creates a flow network $N$ that allows a max flow of value $D 4^{k-1}$ where D is the number of input gates in $C$, iff $C$ evaluates to 1.

The idea is to construct the network by creating two nodes $(i,x,0)$ and $(i,x,1)$ for each gate $(i,x)$ (gate $x$ on the $i$th layer in $C$, one for each possible value 0 and 1). The input layer is layer 0. The source of $N$ is connected to $(0,x,0)$ ($(0,x,1)$) with capacity $4^k$ iff $(0,x)$ is a constant 0 (1) in $C$. Each edge $((i,x),(i+1,y))$ of $C$ is then replaced by a small subnetwork which sends 3/4 of the flow directly to the sink. The remaining quarter is sent to $(i+1,y,0) ((i+1,y,1))$, if $x$ in $C$ values to 0 (1). (At least the max flow needs to take these routes. Hence these small subnetworks simulate the gates.)

Now, the output gate $(k,x)$ is copied as well, but only the $(k,x,1)$ copy is connected to the edge. That is: Iff $C$ evaluates to 0, then exactly 1 flow unit cannot be sent to the sink. Hence, the flow in $N$ is $D 4^{k-1}-1$. If $C$ evaluates to 1, then the full $D 4^{k-1}$ flow units can reach the sink. This completes the reduction.