Orientable domination in product-like graphs

TL;DR

This paper investigates the orientable domination number in various product graphs, providing exact values and bounds, and extends previous results to new classes of graphs, enhancing understanding of domination in directed graph orientations.

Contribution

It determines the orientable domination number for corona, Cartesian, and lexicographic products, and extends known results to multinary complete bipartite graphs, addressing open questions in the field.

Findings

Exact orientable domination number for corona products.

Sharp bounds for Cartesian and lexicographic products.

Counterexample to a previous conjecture on domination and packing numbers.

Abstract

The orientable domination number, ${\rm DOM}(G)$, of a graph $G$ is the largest domination number over all orientations of $G$. In this paper, ${\rm DOM}$ is studied on different product graphs and related graph operations. The orientable domination number of arbitrary corona products is determined, while sharp lower and upper bounds are proved for Cartesian and lexicographic products. A result of Chartrand et al. from 1996 is extended by establishing the values of ${\rm DOM}(K_{n_1,n_2,n_3})$ for arbitrary positive integers $n_1,n_2$ and $n_3$. While considering the orientable domination number of lexicographic product graphs, we answer in the negative a question concerning domination and packing numbers in acyclic digraphs posed in [Domination in digraphs and their direct and Cartesian products, J. Graph Theory 99 (2022) 359-377].