The Domination and Secure Domination Numbers of Direct Product of Cliques with Paths and Cycles

TL;DR

This paper precisely calculates various domination parameters for the direct product of complete graphs with paths or cycles, revealing relationships among these parameters and correcting previous inaccuracies in the literature.

Contribution

It provides exact values for multiple domination parameters in these graph families and clarifies their interrelations, also disproving some earlier incorrect results.

Findings

Independent domination number equals domination number.

Secure domination number equals 2-domination number.

Counterexamples disprove previous erroneous results.

Abstract

In this paper, we obtain the exact values of several domination parameters for the direct product of a complete graph with a path or a cycle. Specifically, we determine the domination number, independent domination number, $[1,2]$-domination number, secure domination number, and 2-domination number for this family of graphs. We show that, in these graphs, the independent domination number and the $[1,2]$-domination number coincide with the domination number, while the secure domination number coincides with the 2-domination number. Additionally, as a consequence of our findings, we provide counterexamples to disprove some erroneous results in the literature.