network flow using minimum number of nodes
The problem is NP-hard on layered graphs. This can be seen by reduction from Exact Cover by 3-sets (X3C). Let $A_1,\ldots,A_m$ be subsets of $\{1,2,\ldots,3n\}$ with $\vert A_i\rvert=3$ for every $i$, the problem asks if there exists a set $I\subseteq\{1,2,\ldots,m\}$ of $\lvert I\rvert=n$ indices such that $\bigcup_{i\in I}A_i=\{1,2,\ldots,3n\}$. This can be reduced to the following instance of your problem. The node set is
$$V=\{s,t\}\cup\{v_i\ :\ i=1,2,\ldots,m\}\cup\{w_j\ :\ j=1,2,\ldots,3n\}.$$
There are the following arcs:


an arc $(s,v_i)$ with capacity 3 for every $i\in\{1,2,\ldots,m\}$,
an arc $(v_i,w_j)$ with capacity 1 for every pair $(i,j)$ with $j\in A_i$
an arc $(w_j,t)$ with capacity 1 for every $j\in\{1,2,\dots,3n\}$.


A flow of value $3n$ can be achieved using $4n$ nodes (not counting $s$ and $t$) if and only if the X3C-instance is a YES-instance.