### The Superior Complement in Graphs

#### Abstract

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 $u\in V(G)$, define $d_D^-(u)=min\{d_D(u,v): v\in V(G)-\{u\}\}$. 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 $A_u=\{v\in V(G): d_D(u,v)\geq d_{D}^-(u)+1\}$. Then $u$ is adjacent to all the vertices $v\in A_u$ 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 $G$.

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