A d-regular graph is said to be superconnected if any disconnecting subset with cardinality at most d is formed by the neighbors of some vertex. A superconnected graph that remains connected after the failure of a vertex and its neighbors will be called vosperian. Let $\Gamma$ be a vertex-transitive graph of degree d with order at least d+4. We give necessary and sufficient conditions for the vosperianity of $\Gamma$. Moreover, assuming that distinct vertices have distinct neighbors, we show that $\Gamma$ is vosperian if and only if it is superconnected. Let G be a group and let S⊂G\{1} with S=$S^{-1}$.We show that the Cayley graph, Cay(G,S), defined on G by S is vosperian if and only if G\(S∪{1}) is not a progression and for every non-trivial subgroup H and every a∈G, |(H∪Ha)(S∪{1})|≥min(|G|−1, |H∪Ha|+|S|+1). If moreover S is aperiodic, then Cay(G,S) is vosperian if and only if it is superconnected.