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arXiv · 2506.11865

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

Deepak M. Bakal·S. A. Mane
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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.

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Deepak M. Bakal, S. A. Mane. 2025-06-13. The Domination and Secure Domination Numbers of Direct Product of Cliques with Paths and Cycles. https://arxiv.org/abs/2506.11865

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