Largest independent sets in triangle free graphs-question about an equality

Question

Let $G=(V,E)$ be a triangle free graph of order $N$ and for $x\in V$, let $D(x)$ be the the set of neighbours of $x$, and $G_x$ be the induced subgraph of $G$ on the subset $V\setminus \{x\}\cup D(x)$. Also, let $\beta(G)$ be the size of the largest independent set of $G$, then according to this notes (end of the first page) $$ \beta(G) = 1+\frac{1}{N}\sum_{x\in V}\beta(G_x). $$ I could not prove the statement. Can someone explain why this equality will be true? (some hints/steps/reference will be very helpful)

Answer 1

Here are the hints. If $M$ is a maximal independent set in $G_x$, then $M\cup\{x\}$ is an independent set in G. Therefore $\beta(G)\geq\beta(G_x)+1$. It follows that $$ \beta(G)\geq\frac{\sum_{x \in V} \beta(G_x)}{N}+1. $$ The converse inequality is wrong, as @AnilCh rightly pointed out.