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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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    $\begingroup$ Not always. For G an odd cycle on more than three vertices, I believe no such independent set E exists. Gerhard "Possibly True For Bipartite Graphs" Paseman, 2018.08.25. $\endgroup$ Aug 26 '18 at 3:47
  • $\begingroup$ just a nitpick - do not use $E$ to denote both edges of $G$ and an independent set. $\endgroup$ Aug 26 '18 at 8:41
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    $\begingroup$ you are also not following the standard in graph theory convention that a triangle is a complete subgraph on 3 vertices. $\endgroup$ Aug 26 '18 at 8:50
  • $\begingroup$ as well, I think, it is important whether $G$ is finite, or not. $\endgroup$ Aug 26 '18 at 8:52
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    $\begingroup$ Possible duplicate of Asking for an example of a graph $G$ satisfying the following property $\endgroup$
    – bof
    Aug 28 '18 at 11:23
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For $n\in\mathbb N$ we have $D(K_2\square K_{n+2})=D((C_{3n+2})^n)=n.$

(I posted these examples in my answer to your other question before I looked at this question.)

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