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A vertex-cut S is called a super vertex-cut if G − S is disconnected and it contains no isolated vertices. The super-connectivity, κ′, is the minimum cardinality over all super vertex-cuts. This article provides bounds for the super connectivity of the direct product of an arbitrary graph and the complete graph K_n. Among other results, we show that if G is a non-complete graph with girth(G) = 3 and κ′(G) = ∞, then κ′(G × K_n) ≤ min{mn − 6, m(n − 1) + 5, 5n + m − 8}, where |V(G)| = m.