Suppose $G$ is a connected finite graph satisfying that every edge $uv$ of $G$ belongs to a "triangle" $uvw$ such that $uv,uw\in E(G),\ vw\notin E(G)$ or $uv,vw\in E(G),\ uw\notin E(G)$(in other words, every edge of $G$ is part of a vee shape), i.e. of an induced $ K_{ 1,2}$.

For any independent set $S$ in $G$ with $|S|\geqslant2$ and $v\in S$, define $$d_S(v)=\mid\{u\in V(G)\setminus S:N_G(u)\cap S=\{v\}\}\mid$$ and $$D_S=\max\{d_S(v):v\in S\},\\D(G)=\min\{D_S:\text{$S$ ranges over all independent sets in $G$ with $|S|\geqslant2$}\}.$$ My question is whether it is possible that for any given integer $n\geq1$, we can always find $G$ such that $D(G)>n$.

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