Is the complexity of this covering problem known? 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.
The problem is known as the propagation problem. Aazami has proved in his PhD thesis that the weighted version is NP-complete even when the graph is planar and the node weights are in $\{0,1\}$. The complexity for the unweighted version seems to be an open problem.