## Šumenjak, Tadeja Kraner and Peterin, Iztok and Rall, Douglas F. and Tepeh, Aleksandra - Partitioning the vertex set of $G$ to make $G\,\Box\, H$ an efficient open domination graph

dmtcs:1503 - Discrete Mathematics & Theoretical Computer Science, June 2, 2016, Vol. 18 no. 3
Partitioning the vertex set of $G$ to make $G\,\Box\, H$ an efficient open domination graph

Authors: Šumenjak, Tadeja Kraner and Peterin, Iztok and Rall, Douglas F. and Tepeh, Aleksandra

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.

Source : oai:arXiv.org:1508.04029
Volume: Vol. 18 no. 3
Section: Graph Theory
Published on: June 2, 2016
Submitted on: June 2, 2016
Keywords: Mathematics - Combinatorics,05C76, 05C69