Geodesic
Partitioning the vertex set of $G$ to make $G\,\Box\, H$ an efficient open domination graph
Discrete mathematics & theoretical computer science, Tome 18 (2015-2016) no. 3.

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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. 
@article{DMTCS_2016_18_3_a9,
     author = {\v{S}umenjak, Tadeja Kraner and Peterin, Iztok and Rall, Douglas F. and Tepeh, Aleksandra},
     title = {Partitioning the vertex set of $G$ to make $G\,\Box\, H$ an efficient open domination graph},
     journal = {Discrete mathematics & theoretical computer science},
     publisher = {mathdoc},
     volume = {18},
     number = {3},
     year = {2015-2016},
     doi = {10.46298/dmtcs.1277},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.46298/dmtcs.1277/}
}
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Šumenjak, Tadeja Kraner; Peterin, Iztok; Rall, Douglas F.; Tepeh, Aleksandra. Partitioning the vertex set of $G$ to make $G\,\Box\, H$ an efficient open domination graph. Discrete mathematics & theoretical computer science, Tome 18 (2015-2016) no. 3. doi : 10.46298/dmtcs.1277. http://geodesic.mathdoc.fr/articles/10.46298/dmtcs.1277/

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