Partitioning the vertex set of $G$ to make $G\,\Box\, H$ an efficient open domination graph

TL;DR

This paper characterizes when the Cartesian product of a graph with certain complete or bipartite graphs results in an efficient open domination graph, based on vertex partition properties of the original graph.

Contribution

It provides a characterization of graphs G for which G □ H is an efficient open domination graph, especially when H is complete or bipartite, including constructive results for trees.

Findings

Characterization of G for which G □ H is an efficient open domination graph

Constructive characterization for trees when H is complete of order at least 3

Identification of efficient open domination graphs when H is a 4-cycle or 5-cycle

Abstract

A graph is an efficient open domination graph if there exists a subset of vertices whose open neighborhoods partition its vertex set. We characterize those graphs $G$ for which the Cartesian product $G \Box H$ is an efficient open domination graph when $H$ is a complete graph of order at least 3 or a complete bipartite graph. The characterization is based on the existence of a certain type of weak partition of $V(G)$. For the class of trees when $H$ is complete of order at least 3, the characterization is constructive. In addition, a special type of efficient open domination graph is characterized among Cartesian products $G \Box H$ when $H$ is a 5-cycle or a 4-cycle.