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

Jovana Forcan, Jiayue Qi

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 $γ_{MB}(G)$ ($γ'_{MB}(G)$) 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 $γ'_{MB}(P_2\square P_n)=n$ for $n\geq 1$, $γ_{MB}(P_2\square P_n)$ 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 $γ_{MB}'(\dot\cup_{i=1}^k(P_2\square P_n)_i)=k\cdot n$ ($n\geq 1$), and that $γ_{MB}(\dot\cup_{i=1}^k(P_2\square P_n)_i)$ 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.

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

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 () of a graph 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 for , equals , , , for , , and , respectively. For the disjoint union of s, we show that (), and that equals , , for , , and , respectively.

Paper Structure

This paper contains 16 sections, 17 theorems, 39 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Background
  3. Preliminaries
  4. Main results
  5. MBD games on the Cartesian product of two graphs
  6. MBD game on $P_2\square P_n$
  7. MBD graphs
  8. Trap 1 - triangle trap.
  9. Trap 2 - line trap.
  10. Maker--Breaker domination numbers of MBD graphs
  11. Domination numbers of MBD graphs: main results
  12. Domination numbers of MBD graphs: proofs
  13. Maker--Breaker domination number of $P_2\square P_n$
  14. MBD game on $P_2\square P_{12}$
  15. Union of grids
  16. ...and 1 more sections

Key Result

Lemma \oldthetheorem

(No-Skip Lemma, GIK19) In an optimal strategy of Dominator to achieve $\gamma_{MB}(G)$ or $\gamma'_{MB}(G)$, it is never an advantage for him to skip a move. Moreover, if Staller skips a move it can never disadvantage Dominator.

Figures (4)

  • Figure 1: Sub-figures: (a) $\mathfrak{X}_m$ (b) $\mathfrak{Y}_m$ (c) $\mathfrak{Z}_m$ (d) $\mathfrak{W}_m$ (e) $\mathfrak{R}_m$. Two incident edges being dotted indicates that the vertex is already dominated by Dominator; the cross indicates that the vertex is claimed by Staller.
  • Figure 2: Illustrations of the two type of traps. (a) Sequence of triangle traps $v_3--v_7$; (b) Sequence of line traps $u_3--u_7$.
  • Figure 3: This is a strategy tree showing the strategies for Staller under the situation when $D_1=u_i(i\geq 3)$, $S_1=v_2$, and $D_3=u_i(1\leq i\leq m)$, in an MBD game on $\mathfrak{X}_m$. At the end of each branch, we explain at least how many steps Dominator would need in order to win the game. It covers the cases (2.1), (2.2) and (2.3) in the proof of Theorem \ref{['thm:x_m']}.
  • Figure 4: This is a strategy tree showing the strategies for Staller under the situation when $D_1=v_6$, in the Maker--Breaker domination game on $P_2\square P_{12}$. One can verify that with the provided strategy for Staller, Dominator always needs at least $11$ steps to win.

Theorems & Definitions (48)

  • Remark \oldthetheorem
  • Lemma \oldthetheorem
  • Theorem \oldthetheorem
  • Theorem \oldthetheorem
  • Theorem \oldthetheorem
  • Theorem \oldthetheorem
  • Theorem \oldthetheorem
  • proof : of Theorem \ref{['thm:bound1']}
  • Corollary \oldthetheorem
  • Proposition \oldthetheorem
  • ...and 38 more