On the $n-$dominating graph of $K_{n}$

Saeid Alikhani, Davood Fatehi

Published 2014-10-05Version 1

Let $G=(V,E)$ be a graph. A set $S\subseteq V(G)$ is a dominating set, if every vertex in $V(G)\backslash S$ is adjacent to at least one vertex in $S$. The $k$-dominating graph of $G$, $D_k (G)$, is defined to be the graph whose vertices correspond to the dominating sets of $G$ that have cardinality at most $k$. Two vertices in $D_k(G)$ are adjacent if and only if the corresponding dominating sets of $G$ differ by either adding or deleting a single vertex. In this paper we consider $D_n(K_n)$ and study some of its properties.