The K1,2-structure-connectivity of graphs

Abstract

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