Efficient closed domination in digraph products

TL;DR

This paper characterizes efficient closed domination digraphs in various digraph products, providing complete descriptions for lexicographic, strong, and certain direct and Cartesian products.

Contribution

It offers a comprehensive characterization of efficient closed domination in multiple digraph product types, including lexicographic, strong, direct, and Cartesian products.

Findings

Complete description of efficient closed domination in lexicographic and strong products.

Characterization of direct products with cycles or paths as efficient closed domination digraphs.

Description of Cartesian product digraphs with cycles or stars that are efficient closed domination digraphs.

Abstract

A digraph $D$ is an efficient closed domination digraph if there exists a subset $S$ of $V(D)$ for which the closed out-neighborhoods centered in vertices of $S$ form a partition of $V(D)$. In this work we deal with efficient closed domination digraphs among several product of digraphs. We completely describe the efficient closed domination digraphs among lexicographic and strong products of digraphs. We characterize those direct products of digraphs that are efficient closed domination digraphs, where factors are either two cycles or two paths. Among Cartesian product of digraphs, we describe all such efficient closed domination digraphs such that they are a Cartesian product digraph either with a cycle or with a star.