On the independence number in subcubic graphs

Abstract

For a connected subcubic graph $G\ne K_1$ let $V_i(G)=\{v\in V(G)|d_G(v)=i\}$ for $1\le i\le 3.$ Given $c_1,c_2,c_3\in {\mathbb{R}}^{+}$ and $d\in \mathbb{R}$, we show several results of type $\hat{I}\pm (G)\ge c_1|V_1(G)|+c_2|V_2(G)|+c_3|V_3(G)|-d.$ We also derive classes of graphs $G$ showing sharpness of these lower bounds on the independence number $\hat{I}\pm (G)$ of $G$.