Maker-Breaker domination number for Cartesian products of path graphs $P_2$ and $P_n$

TL;DR

This paper analyzes the Maker-Breaker domination game on Cartesian products of path graphs, providing exact values and bounds for the domination numbers, especially for the product of $P_2$ and $P_n$, and extends results to disjoint unions.

Contribution

It establishes exact values and bounds for the Maker-Breaker domination number on Cartesian products of paths, including new formulas for $P_2 imes P_n$ and their disjoint unions.

Findings

$oldsymbol{ ext{For } P_2 imes P_n, ext{ the second-player domination number is } n}$.

$oldsymbol{ ext{For } P_2 imes P_n, ext{ the first-player domination number varies with } n$ (exact for small and large } n).

$oldsymbol{ ext{Bounds are provided for Cartesian products involving arbitrary graphs and complete graphs.}}

Abstract

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.