Revisiting $k$-tuple dominating sets with emphasis on small values of $k$

TL;DR

This paper introduces the double Slater number as a new lower bound for double domination, explores its computational complexity, and characterizes specific graph families, including full graphs, to advance understanding of $k$-tuple domination.

Contribution

It defines the double Slater number, proves its relation to double domination, analyzes the complexity of related decision problems, and characterizes full graphs.

Findings

Double Slater number bounds double domination number.

Deciding equality of these parameters is NP-complete for 4-partite graphs.

Computing double domination is NP-hard for comparability graphs of diameter two.

Abstract

For any graph $G$ of order $n$ with degree sequence $d_{1}\geq\cdots\geq d_{n}$, we define the double Slater number $s\ell_{\times2}(G)$ as the smallest integer $t$ such that $t+d_{1}+\cdots+d_{t-e}\geq2n-p$ in which $e$ and $p$ are the number of end-vertices and penultimate vertices of $G$, respectively. We show that $\gamma_{\times2}(G)\geq s\ell_{\times2}(G)$, where $\gamma_{\times2}(G)$ is the well-known double domination number of a graph $G$ with no isolated vertices. We prove that the problem of deciding whether the equality holds for a given graph is NP-complete even when restricted to $4$-partite graphs. We also prove that the problem of computing $\gamma_{\times2}(G)$ in NP-hard even for comparability graphs of diameter two. Some results concerning these two parameters are given in this paper improving and generalizing some earlier results on double domination in graphs. We give an upper bound on the $k$-tuple domatic number of graphs with characterization of all graphs attaining the bound. Finally, we characterize the family of all full graphs, leading to a solution to an open problem given in a paper by Cockayne and Hedetniemi ($1977$).