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- W3048184629 abstract "A set $D$ of vertices in a graph $G$ is called dominating if every vertex of $G$ is either in $D$ or adjacent to a vertex of $D$. The paired domination number $gamma_{mathrm{pr}}(G)$ of $G$ is the minimum size of a dominating set whose induced subgraph admits a perfect matching, and the upper domination number $Gamma(G)$ is the maximum size of a minimal dominating set. In this paper, we investigate the sharpness of two multiplicative inequalities for these domination parameters, where the graph product is the direct product $times$. We show that for every positive constant $c$, there exist graphs $G$ and $H$ of arbitrarily large diameter such that $gamma_{mathrm{pr}}(G times H) leq cgamma_{mathrm{pr}}(G)gamma_{mathrm{pr}}(H)$, thus answering a question of Rall as well as two questions of Paulraja and Sampath Kumar. We then study when this inequality holds with $c = frac{1}{2}$, in particular proving that it holds whenever $G$ and $H$ are trees. Finally, we demonstrate that the inequality $Gamma(G times H) geq Gamma(G) Gamma(H)$, due to Bresar, Klavžar, and Rall, is tight." @default.
- W3048184629 created "2020-08-13" @default.
- W3048184629 creator A5024470947 @default.
- W3048184629 date "2020-08-10" @default.
- W3048184629 modified "2023-09-27" @default.
- W3048184629 title "Tightness of Paired and Upper Domination Inequalities for Direct Product Graphs" @default.
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