Extremal Graphs for the \(K_{1,2}\)-Isolation Number of Graphs

For any non-negative integer k and any graph G, a subset \(S\subseteq V(G)\) $S\subseteq V(G)$ is said to be a \(K_{1,k+1}\) $K_{1,k+1}$ -isolating set of G if \(G-N[S]\) $G-N[S]$ does not contain \(K_{1,k+1}\) $K_{1,k+1}$ as a subgraph. The \(K_{1,k+1}\) $K_{1,k+1}$ -isolation number of G, denoted by \(\iota _k(G)\) ${\iota }_k(G)$ , is the minimum cardinality of a \(K_{1,k+1}\) $K_{1,k+1}$ -isolating set of G. Recently, Zhang and Wu (2021) proved that if G is a connected n-vertex graph and \(G\notin \{P_3,C_3,C_6\}\) $G\notin \{P_3,C_3,C_6\}$ , then \(\iota _1(G)\le \frac{2}{7}n\) ${\iota }_1(G)\le \frac{2}{7}n$ . 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).