The characterization of 2-quasi-regularizable connected W_2 graphs

Let $G$ be a connected graph in $\boldsymbol{W}_2$, and let $n(G)$ denote its number of vertices and $\alpha(G)$ its independence number. The graph $G$ is $2$-quasi-regularizable if

$$2\lvert S\rvert\leq\lvert N_G(S)\rvert$$

for every independent set $S$ of $G$.

The 2-quasi-regularizability characterization. $G$ is $2$-quasi-regularizable if and only if

$$n(G)\geq3\alpha(G).$$

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