The K_1,2-structure-connectivity of graphs

In this paper, we focus on examining the \(K_{1,2}\) $K_{1,2}$ -structure-connectivity of any connected graph. Let G be a connected graph with n vertices, we show that \(\kappa (G; K_{1,2})\) $\kappa (G;K_{1,2})$ is well defined if \(\hbox {diam}(G)\ge 4\) $\text{diam}(G)\ge 4$ , or \(n\equiv 1\pmod 3\) $n\equiv 1(mod3)$ , or \(G\notin \{C_{5},K_{n}\}\) $G\notin \{C_5,K_n\}$ when \(n\equiv 2\pmod 3\) $n\equiv 2(mod3)$ , or there exist three vertices u, v, w such that \(N_{G}(u)\cap (N_{G}(\{v,w\})\cup \{v,w\})=\emptyset\) $N_G(u)\cap (N_G(\{v,w\})\cup \{v,w\})=\mathrm{\varnothing }$ when \(n\equiv 0\pmod 3\) $n\equiv 0(mod3)$ . Furthermore, if G has \(K_{1,2}\) $K_{1,2}$ -structure-cut, we prove \(\kappa (G)/3\le \kappa (G; K_{1,2})\le \kappa (G)\) $\kappa (G)/3\le \kappa (G;K_{1,2})\le \kappa (G)$ .