On the independence number in subcubic graphs

For a connected subcubic graph $G\mathrm{\ne }{K}_1$ let $V_i(G)=\{v\in V(G)\mathrm{\mid }{d}_G(v)=i\}$ 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 $\alpha (G)\ge c_1\mathrm{\mid }{V}_1(G)\mathrm{\mid }+c_2\mathrm{\mid }{V}_2(G)\mathrm{\mid }+c_3\mathrm{\mid }{V}_3(G)\mathrm{\mid }-d\mathrm{.}$ We also derive classes of graphs $G$ showing sharpness of these lower bounds on the independence number $\alpha (G)$ of $G$.