Extremal Graphs for the $$K_{1,2}$$ -Isolation Number of Graphs

For any non-negative integer k and any graph G, a subset \(S\subseteq V(G)\) 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 \(\iota _k(G)\), 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 \(G\notin \{P_3,C_3,C_6\}\), then \(\iota _1(G)\le \frac{2}{7}n\). 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).