We first define the trace on a domain $\Omega$ which is definable in an o-minimal structure. then show that every function $u\in W^{1,p}(\Omega)$ vanishing boundary sense satisfies Poincar\'e inequality. finally show, given family of domains $(\Omega_t)_{t\in \mathbb{R}^k}$, constant this inequality remains bounded, if so does volume $\Omega_t$.

we define minimal cn-dominating graph $mathbf {mcn}(g)$, commonality minimal cn-dominating graph $mathbf {cmcn}(g)$ and vertex minimal cn-dominating graph $mathbf {m_{v}cn}(g)$, characterizations are given for graph $g$ for which the newly defined graphs are connected. further serval new results are developed relating to these graphs.