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Thresholds for the biased Maker-Breaker domination games

Boštjan Brešar, Csilla Bujtás, Pakanun Dokyeesun, Tanja Dravec

TL;DR

This work investigates biased Maker-Breaker domination games on graphs, focusing on thresholds ${\rm a}_b(G)$, ${\rm a}'_b(G)$, ${\rm b}_a(G)$, and ${\rm b}'_a(G)$ that determine winner outcomes under optimal play when players claim multiple vertices per move. It introduces the $\ell$-local domination number $\widetilde{\gamma}_\ell(G)$ and the star partition width $\sigma(G)$ as key tools for bounding and computing thresholds, and provides a suite of results across graph families: large-order graphs (where Staller often wins for certain biases), line graphs (with exact thresholds under degree conditions), grids (finite and infinite) with several exact or tight bounds, and trees via equality ${\rm a}_1'(T)=\sigma(T)$. The paper also proves structural results for line graphs using SDR$^t$ representations and Hall's theorem, and develops a nuanced analysis of grids, including modular cases when $m\equiv1\pmod4$. Together, these findings connect graph structure to domination strategies and establish several sharp bounds and exact values, while outlining open problems for grids and sparse graph classes.

Abstract

In the $(a,b)$-biased Maker-Breaker domination game, two players alternately select unplayed vertices in a graph $G$ such that Dominator selects $a$ and Staller selects $b$ vertices per move. Dominator wins if the vertices he selected during the game form a dominating set of $G$, while Staller wins if she can prevent Dominator from achieving this goal. Given a positive integer $b$, Dominator's threshold, $\textrm{a}_b$, is the minimum $a$ such that Dominator wins the $(a,b)$-biased game on $G$ when he starts the game. Similarly, $\textrm{a}'_b$ denotes the minimum $a$ such that Dominator wins when Staller starts the $(a,b)$-biased game. Staller's thresholds, $\textrm{b}_a$ and $\textrm{b}'_a$, are defined analogously. It is proved that Staller wins the $(k-1,k)$-biased games in a graph $G$ if its order is sufficiently large with respect to a function of $k$ and the maximum degree of $G$. Along the way, the $\ell$-local domination number of a graph is introduced. This new parameter is proved to bound Dominator's thresholds $\textrm{a}_\ell$ and $\textrm{a}_\ell'$ from above. As a consequence, $\textrm{a}_1'(G)\le 2$ holds for every claw-free graph $G$. More specific results are obtained for thresholds in line graphs and Cartesian grids. Based on the concept of $[1,k]$-factor of a graph $G$, we introduce the star partition width $σ(G)$ of $G$, and prove that $\textrm{a}_1'(G)\le σ(G)$ holds for any nontrivial graph $G$, while $\textrm{a}_1'(G)=σ(G)$ if $G$ is a tree.

Thresholds for the biased Maker-Breaker domination games

TL;DR

This work investigates biased Maker-Breaker domination games on graphs, focusing on thresholds , , , and that determine winner outcomes under optimal play when players claim multiple vertices per move. It introduces the -local domination number and the star partition width as key tools for bounding and computing thresholds, and provides a suite of results across graph families: large-order graphs (where Staller often wins for certain biases), line graphs (with exact thresholds under degree conditions), grids (finite and infinite) with several exact or tight bounds, and trees via equality . The paper also proves structural results for line graphs using SDR representations and Hall's theorem, and develops a nuanced analysis of grids, including modular cases when . Together, these findings connect graph structure to domination strategies and establish several sharp bounds and exact values, while outlining open problems for grids and sparse graph classes.

Abstract

In the -biased Maker-Breaker domination game, two players alternately select unplayed vertices in a graph such that Dominator selects and Staller selects vertices per move. Dominator wins if the vertices he selected during the game form a dominating set of , while Staller wins if she can prevent Dominator from achieving this goal. Given a positive integer , Dominator's threshold, , is the minimum such that Dominator wins the -biased game on when he starts the game. Similarly, denotes the minimum such that Dominator wins when Staller starts the -biased game. Staller's thresholds, and , are defined analogously. It is proved that Staller wins the -biased games in a graph if its order is sufficiently large with respect to a function of and the maximum degree of . Along the way, the -local domination number of a graph is introduced. This new parameter is proved to bound Dominator's thresholds and from above. As a consequence, holds for every claw-free graph . More specific results are obtained for thresholds in line graphs and Cartesian grids. Based on the concept of -factor of a graph , we introduce the star partition width of , and prove that holds for any nontrivial graph , while if is a tree.

Paper Structure

This paper contains 13 sections, 19 theorems, 11 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Structure of the paper.
  3. Standard definitions.
  4. Preliminaries
  5. Graphs of large order
  6. S($1$)
  7. S($i$)
  8. Local domination and an upper bound on ${\rm a}_\ell'(G)$
  9. Thresholds for line graphs
  10. Claim 11.A.
  11. A summary of thresholds for grids
  12. Star partition width and ${\rm a}_1'(G)$
  13. Concluding remarks

Key Result

Proposition 2

The following equivalences are true for every graph $G$ and positive integers $i$ and $j$.

Figures (3)

  • Figure 1: The graph $G_{4,3}$.
  • Figure 2: A graph $G$.
  • Figure 3: A graph $G$ with star partition ${\cal{S}}=\{S_1,S_2,S_3\}$ and its star digraph $H_{G,{\cal{S}}}$

Theorems & Definitions (34)

  • Proposition 2
  • Proposition 3
  • proof
  • Remark 4
  • Theorem 5
  • proof
  • Remark 6
  • Remark 7
  • Theorem 8
  • proof
  • ...and 24 more