Summary: 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 \(S \subset G \setminus \{1\}\) with \(S=S^{-1}\). We show that the Cayley graph, \(Cay(G, S)\), defined on \(G\) by \(S\) is vosperian if and only if \(G \setminus (S \cup \{1\})\) is not a progression and for every non-trivial subgroup \(H\) and every \(a \in G\),
\[ \left|(H \cup Ha)(S \cup \{1\})\right| \min (|G| -1, |H \cup Ha| + |S| + q). \] If moreover \(S\) is aperiodic, then \(Cay(G, S)\) is vosperian if and only if it is superconnected.