The super-connectivity of the double vertex graph of complete bipartite graphs

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Let $ G=(V,E) $ be a graph. The double vertex graph $F_2(G)$ of $ G $ is the graph whose vertex set consists of all $ 2 $-subsets of $ V(G) $ such that two vertices are adjacent in $ F_2(G) $ if their symmetric difference is a pair of adjacent vertices in $ G $. The super--connectivity of a connected graph is the minimum number of vertices whose removal results in a disconnected graph without an isolated vertex. In this paper, we determine the super--connectivity of the double vertex graph of the complete bipartite graph $ K_{m,n} $ for $ m\geq 4 $ where $ n\geq m+2 $.