Domination related parameters in the generalized lexicographic product of graphs

TL;DR

This paper investigates various domination-related parameters in the generalized lexicographic product of graphs, establishing equality chains and extending known results, including characterizations for well-dominated cases.

Contribution

It introduces new equality chains for domination parameters in the generalized lexicographic product and extends existing results to broader classes of domination parameters.

Findings

Established equality chains for domination parameters in GLP

Extended known results from standard to generalized lexicographic product

Characterized well μ-dominated GLP for domination and total domination numbers

Abstract

In this paper we begin an exploration of several domination-related parameters (among which are the total, restrained, total restrained, paired, outer connected and total outer connected domination numbers) in the generalized lexicographic product (GLP for short) of graphs. We prove that for each GLP of graphs there exist several equality chains containing these parameters. Some known results on standard lexicographic product of two graphs are generalized or/and extended. We also obtain results on well $\mu$-dominated GLP of graphs, where $\mu$ stands for any of the above mentioned domination parameters. In particular, we present a characterization of well $\mu$-dominated GLP of graphs in the cases when $\mu$ is the domination number or the total domination number.