The $K_{1,2}$-structure-connectivity of graphs

Abstract

In this paper, we mainly investigate $K_{1,2}$-structure-connectivity for any connected graph. Let $G$ be a connected graph with $n$ vertices, we show that $κ(G; K_{1,2})$ is well-defined if $diam(G)\geq 4$, 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 $κ(G)/3\leqκ(G; K_{1,2})\leqκ(G)$.

Figures (15)

• Figure 1: The illustration of Case 1
• Figure 2: $\{wy_1x_1\}$ is a $K_{1,2}$-structure-cut of $G-{\cal F}$
• Figure 3: $\{x_1ux_2,y_1wv\}$ is a $K_{1,2}$-structure-cut of $G-{\cal F}$
• Figure 4: $\{vx_iw,x_1ux_2\}$ is a $K_{1,2}$-structure-cut of $G-{\cal F}$
• Figure 5: $\{x_1y_1w,x_ivy_3\}$ is a $K_{1,2}$-structure-cut of $G-{\cal F}$

Theorems & Definitions (8)

• Theorem 3.1
• Lemma 3.2
• Lemma 3.3
• Lemma 3.4
• Remark 3.5
• Theorem 3.6
• Lemma 4.1
• Theorem 4.2