Thresholds for the biased Maker-Breaker domination games

Brešar, Boštjan (author), Bujtás, Csilla (author), Dokyeesun, Pakanun (author), Dravec, Tanja (author)

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, ${\rm a}_b$, is the minimum $a$ such that Dominator wins the $(a,b)$-biased game on $G$ when he starts the game. Similarly, ${\rm a}'_b$ denotes the minimum $a$ such that Dominator wins when Staller starts the $(a,b)$-biased game. Staller's thresholds, ${\rm b}_a$ and ${\rm b}'_a$, 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 ${\rm a}_\ell$ and ${\rm a}_\ell'$ from above. As a consequence, ${\rm a}_1'(G)\le 2$ 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 $\sigma(G)$ of $G$, and prove that ${\rm a}_1'(G)\le \sigma(G)$ holds for any nontrivial graph $G$, while ${\rm a}_1'(G)=\sigma(G)$ if $G$ is a tree.

• Language: English
• Keywords: Maker-Breaker domination game, biased Maker-Breaker game, trees, line graph, grid
• Work type: Article
• Typology: 1.01 - Original Scientific Article
• Organization: FMF - Faculty of Mathematics and Physics
• Publication status: Published
• Publication version: Version of Record
• Publication date: 01.01.2025
• Year: 2025
• Number of pages: 19 str.
• Numbering: Vol. 27, no. 3, article no. 19
• PID: 20.500.12556/RUL-175289
• UDC: 519.17
• ISSN on article: 1365-8050
• DOI: 10.46298/dmtcs.15392
• COBISS.SI-ID: 254459907
• Publication date in RUL: 23.10.2025

Record is a part of a journal

• Title: Discrete mathematics & theoretical computer science
• Shortened title: Discret. math. theor. comput. sci.
• Publisher: DMTCS
• ISSN: 1365-8050
• COBISS.SI-ID: 8089433

Licences

• License: CC BY-NC-SA 4.0, Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International
• Link: http://creativecommons.org/licenses/by-nc-sa/4.0/
• Description: A Creative Commons license that bans commercial use and requires the user to release any modified works under this license.

Secondary language

• Language: Slovenian
• Keywords: dominacijska igra izdelovalec-lomilec, pristranska igra izdelovalec-lomilec, drevesa, graf povezav, mreža