Extremal Graphs for the K1,2-Isolation Number of Graphs.

For any non-negative integer k and any graph G, a subset S ⊆ V (G) is said to be a K 1 , k + 1 -isolating set of G if G - N [ S ] does not contain K 1 , k + 1 as a subgraph. The K 1 , k + 1 -isolation number of G, denoted by ι k (G) , is the minimum cardinality of a K 1 , k + 1 -isolating set of G. Recently, Zhang and Wu (2021) proved that if G is a connected n-vertex graph and G ∉ { P 3 , C 3 , C 6 } , then ι 1 (G) ≤ 2 7 n . In this paper, we characterize all extremal graphs attaining this bound, which resolves a problem proposed by Zhang and Wu (Discrete Appl Math 304:365–374, 2021). [ABSTRACT FROM AUTHOR]