Let $G=(V,E)$ be a graph. A vertex set $X\subseteq V$ is called critical if $X\neq\emptyset$ and no vertex in $V\setminus X$ is adjacent to exactly one vertex in $X$. The problem is to find a vertex set $S\subseteq V$ of minimum size such that $S\cap X\neq\emptyset$ for every critical set $X$.
The problem has the following rumour-spreading interpretation: Vertex $i$ spreads the rumour to its neighbour $j$ if and only if all other neighbours of $i$ are already informed. The question is then how many vertices do I have to inform initially to make sure that everybody is informed in the end.