A zero-sum problem on graphs

Call a graph $G$ zero-forcing for a finite abelian group $\mathcal{G}$ if for every $\mathrm{\ell }:V(G)\to \mathcal{G}$ there is a connected $A\subseteq V(G)$ with ${\sum }_{a\in A}\mathrm{\ell }(a)=0$. 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 $\mathrm{\mid }\mathcal{G}\mathrm{\mid }$ 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.