Inviting an optimal subset of persons such that all friends of a person invited are invited
This is known as the closure problem. I quote the Wikipedia paragraph describing the reduction to the max flow problem here.

As Picard (1976) [1] showed, a maximum-weight closure may be obtained from $G$ by solving a maximum flow problem on a graph $H$ constructed from $G$ by adding to it two additional vertices $s$ and $t$. For each vertex $v$ with positive weight in $G$, the augmented graph $H$ contains an edge from $s$ to $v$ with capacity equal to the weight of $v$, and for each vertex $v$ with negative weight in $G$, the augmented graph $H$ contains an edge from $v$ to $t$ whose capacity is the negation of the weight of $v$. All of the edges in $G$ are given infinite capacity in $H$.
A minimum cut separating $s$ from $t$ in this graph cannot have any edges of $G$ passing in the forward direction across the cut: a cut with such an edge would have infinite capacity and would not be minimum. Therefore, the set of vertices on the same side of the cut as $s$ automatically forms a closure $C$. The capacity of the cut equals the weight of all positive-weight vertices minus the weight of the vertices in $C$, which is minimized when the weight of $C$ is maximized. By the max-flow min-cut theorem, a minimum cut, and the optimal closure derived from it, can be found by solving a maximum flow problem.


[1]  Picard, Jean-Claude (1976), &quot;Maximal closure of a graph and applications to combinatorial problems&quot;, Management Science, 22 (11): 1268–1272, doi:10.1287/mnsc.22.11.1268, MR 0403596.