The Extended Dominating Sets in Graphs

Abstract

Let [Formula: see text] be a graph and let [Formula: see text] be an integer. A vertex subset [Formula: see text] is called a [Formula: see text]-extended dominating set if every vertex [Formula: see text] of [Formula: see text] satisfies one of the following conditions: the distance between [Formula: see text] and [Formula: see text] is at most one or there are at least [Formula: see text] different vertices [Formula: see text] such that the distance between [Formula: see text] and [Formula: see text] ([Formula: see text]) is two. The [Formula: see text]-extended domination number [Formula: see text] of [Formula: see text] is the minimum size over all [Formula: see text]-extended dominating sets in [Formula: see text]. When [Formula: see text], they are called the extended dominating set and the extended domination number of [Formula: see text], respectively. In this paper, we mainly study the bounds of the extended domination numbers of graphs. First, we obtain the exact values of the extended domination numbers for paths and cycles. And then the Nordhaus–Gaddum bounds for the extended domination numbers are provided. Additionally, we give some bounds of the extended domination numbers for planar graphs with small diameters. Finally, we consider the [Formula: see text]-extended domination numbers in Random graphs.