Protection of graphs with emphasis on Cartesian product graphs

TL;DR

This paper investigates domination parameters in graphs, especially Cartesian product graphs, establishing bounds, inequalities, and focusing on weak Roman and secure domination numbers.

Contribution

It provides new bounds and Nordhaus-Gaddum inequalities for secure and weak Roman domination numbers, with specific analysis on Cartesian product graphs.

Findings

Derived general bounds for weak Roman and secure domination numbers.

Established new Nordhaus-Gaddum inequalities involving these parameters.

Analyzed the case of Cartesian product graphs in detail.

Abstract

In this paper we study the weak Roman domination number and the secure domination number of a graph. In particular, we obtain general bounds on these two parameters and, as a consequence of the study, we derive new inequalities of Nordhaus-Gaddum type involving secure domination and weak Roman domination. Furthermore, the particular case of Cartesian product graphs is considered.