Isolation of connected graphs

10/07/2021

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by   Peter Borg, et al.

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For a connected n-vertex graph G and a 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 contains no graph in ℱ. Let ℰ_k denote the set of connected graphs that have at least k edges. By a result of Caro and Hansberg, ι(G,ℰ_1) ≤ n/3 if n ≠ 2 and G is not a 5-cycle. The author recently showed that if G is not a triangle and 𝒞 is the set of cycles, then ι(G,𝒞) ≤ n/4. We improve this result by showing that ι(G,ℰ_3) ≤ n/4 if G is neither a triangle nor a 7-cycle. Let r be the number of vertices of G that have only one neighbour. We determine a set 𝒮 of six graphs such that ι(G,ℰ_2) ≤ (4n - r)/14 if G is not a copy of a member of 𝒮. The bounds are sharp.