## Jovana Forcan ; Jiayue Qi - Maker-Breaker domination number for Cartesian products of path graphs $P_2$ and $P_n$

dmtcs:10465 - Discrete Mathematics & Theoretical Computer Science, April 12, 2024, vol. 25:2 - https://doi.org/10.46298/dmtcs.10465
Maker-Breaker domination number for Cartesian products of path graphs $P_2$ and $P_n$Article

Authors: Jovana Forcan ; Jiayue Qi

We study the Maker-Breaker domination game played by Dominator and Staller on the vertex set of a given graph. Dominator wins when the vertices he has claimed form a dominating set of the graph. Staller wins if she makes it impossible for Dominator to win, or equivalently, she is able to claim some vertex and all its neighbours. Maker-Breaker domination number $\gamma_{MB}(G)$ ($\gamma '_{MB}(G)$) of a graph $G$ is defined to be the minimum number of moves for Dominator to guarantee his winning when he plays first (second). We investigate these two invariants for the Cartesian product of any two graphs. We obtain upper bounds for the Maker-Breaker domination number of the Cartesian product of two arbitrary graphs. Also, we give upper bounds for the Maker-Breaker domination number of the Cartesian product of the complete graph with two vertices and an arbitrary graph. Most importantly, we prove that $\gamma'_{MB}(P_2\square P_n)=n$ for $n\geq 1$, $\gamma_{MB}(P_2\square P_n)$ equals $n$, $n-1$, $n-2$, for $1\leq n\leq 4$, $5\leq n\leq 12$, and $n\geq 13$, respectively. For the disjoint union of $P_2\square P_n$s, we show that $\gamma_{MB}'(\dot\cup_{i=1}^k(P_2\square P_n)_i)=k\cdot n$ ($n\geq 1$), and that $\gamma_{MB}(\dot\cup_{i=1}^k(P_2\square P_n)_i)$ equals $k\cdot n$, $k\cdot n-1$, $k\cdot n-2$ for $1\leq n\leq 4$, $5\leq n\leq 12$, and $n\geq 13$, respectively.

Volume: vol. 25:2
Section: Graph Theory
Published on: April 12, 2024
Accepted on: October 22, 2023
Submitted on: December 13, 2022
Keywords: Mathematics - Combinatorics,91A24, 05C69, 05C57