# On Cartesian products having a minimum dominating set that is a box or a stairway

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Douglas Rall

Furman University

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Bo\v{s}tjan Bre\v{s}ar

University of Maribor \& \\ Institute of Mathematics, Physics and Mechanics

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**Minisymposium:**
GRAPH PRODUCTS

**Content:**
Combining a lower bound due to El-Zahar and Pareek with an upper bound of Vizing we get that for any pair of graphs $G$ and $H$,
\[ \min\{|V(G)|,|V(H)|\}\le \gamma(G\,\Box\, H) \le \min \{\gamma(G)|V(H)|, \gamma(H)|V(G)|\}\,.\]
We give a complete characterization of the pairs that achieve the lower bound and give some necessary and some sufficient conditions for pairs of graphs that meet the upper bound.