Isolation of regular graphs, stars and k-chromatic graphs
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For any graph G and any set ℱ of graphs, let ι(G,ℱ) 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 does not contain a copy of a graph in ℱ. Thus, ι(G,{K_1}) is the domination number of G. For any integer k ≥ 1, let ℱ_0,k = {K_1,k}, let ℱ_1,k be the set of regular graphs of degree at least k-1, let ℱ_2,k be the set of graphs whose chromatic number is at least k, and let ℱ_3,k be the union of ℱ_0,k, ℱ_1,k and ℱ_2,k. We prove that if G is a connected n-vertex graph and ℱ = ℱ_0,k∪ℱ_1,k, then ι(G, ℱ) ≤n/k+1 unless G is a k-clique or k = 2 and G is a 5-cycle. This generalizes a classical bound of Ore on the domination number, a bound of Caro and Hansberg on the {K_1,k}-isolation number, a bound of the author on the cycle isolation number, and a bound of Fenech, Kaemawichanurat and the author on the k-clique isolation number. By Brooks' Theorem, the same holds if ℱ = ℱ_3,k. The bounds are sharp.
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