Nordhaus-Gaddum type inequalities for multiple domination and packing parameters in graphs

TL;DR

This paper explores Nordhaus-Gaddum type inequalities for various domination and packing parameters in graphs, providing new bounds and characterizations for these graph invariants and their complements.

Contribution

It introduces new upper bounds for domination and packing numbers, including stronger bounds for specific cases, and characterizes graphs that attain these bounds.

Findings

Established new upper bounds for domination and packing parameters.

Characterized graphs that attain the bounds for these parameters.

Provided stronger bounds for the sum of domination numbers in graphs and their complements.

Abstract

We study the Nordhaus-Gaddum type results for $(k-1,k,j)$ and $k$-domination numbers of a graph $G$ and investigate these bounds for the $k$-limited packing and $k$-total limited packing numbers in graphs. As the special case $(k-1,k,j)=(1,2,0)$ we give an upper bound on $dd(G)+dd(\overline{G})$ stronger than that presented by Harary and Haynes (1996). Moreover, we establish upper bounds on the sum and product of packing and open packing numbers and characterize all graphs attaining these bounds.