The superior complement in graphs

For distinct vertices $u$ and $v$ of a nontrivial connected graph $G$, we let $D_{u,v}=N[u]cup N[v]$. We define a $D_{u,v}$-walk as a $u$-$v$ walk in $G$ that contains every vertex of $D_{u,v}$. The superior distance $d_D(u,v)$ from $u$ to $v$ is the length of a shortest $D_{u,v}$-walk. For each vertex $uin V(G)$, define $d_D^-(u)=min{d_D(u,v): vin V(G)-{u}}$. A vertex $v(neq u)$ is called a {it superior neighbor} of $u$ if $d_D(u,v)=d_{D}^-(u)$. In this paper we define the concept of superior complement of a graph $G$ as follows: The superior complement of a graph $G$ is denoted by $overline{G}_D$ whose vertex set is as in $G$. For a vertex $u$, let $A_u={vin V(G): d_D(u,v)geq d_{D}^-(u)+1}$. Then $u$ is adjacent to all the vertices $vin A_u$ in $overline{G}_D$. The main focus of this paper is to prove that there is no relationship between the superior diameter $d_D(G)$ of a graph $G$ and the superior diameter $d_D(overline{G}_D)$ of the superior complement $overline{G}_D$ of $G$.