Isolation of k-cliques
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For any positive integer k and any n-vertex graph G, let ι(G,k) denote the size of a smallest set D of vertices of G such that the graph obtained from G by deleting the closed neighbourhood of D contains no k-clique. Thus, ι(G,1) is the domination number of G. We prove that if G is connected, then ι(G,k) ≤n/k+1 unless G is a k-clique or k = 2 and G is a 5-cycle. The bound is sharp. The case k=1 is a classical result of Ore, and the case k=2 is a recent result of Caro and Hansberg. Our result solves a problem of Caro and Hansberg.
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