This question comes from another question I submitted earlier.

Let $G$ be a finite graph. 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 $$D_S=\max\{d_S(v):v\in S\},\\D(G)=\min\{D_S:\text{$S$ ranges over all independent sets in $G$ with $|S|\geqslant2$}\}.$$ We call $G$ a $T$-graph if $G$ 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).

From a comment in this question, I know that if $G$ is an odd cycle on more than three vertices, then $G$ is a $T$-graph with $D(G)=1$. But I can not find a $T$-graph $G$ with $D(G)\geqslant2$, so I submit this question to ask for an example of such $G$.