. Let $S$ and $S'$ be two subdivided stars. We say that $S$ and $S'$ are $\sim_0$-equivalent, denoted $S \sim_0 S'$, if and only if for any subdivided star $\hat{S}$, $S{{\tikz[baseline=-0]{\draw (0,0) node[circle,fill=black,minimum size=0,inner sep=0]{} -- (0.0,0) node[circle,fill=black, minimum size=0, inner sep=0] {} node[midway,above,scale=0.0] {$0$};}}}\hat{S}