  Call a graph $G$ zero-forcing for a finite abelian group $\mathcal{G}$ if for
every $\ell : V(G) \to \mathcal{G}$ there is a connected $A \subseteq V(G)$
with $\sum_{a \in A} \ell(a) = 0$. The problem we pose here is to characterise
the class of zero-forcing graphs. It is shown that a connected graph is
zero-forcing for the cyclic group of prime order $p$ if and only if it has at
least $p$ vertices. When $|\mathcal{G}|$ is not prime, however, being
zero-forcing is intimately linked to the structure of the graph. We obtain
partial solutions for the general case, discuss computational issues and
present several questions.
