Indicated domination game

Brešar, Boštjan (author), Bujtás, Csilla (author), Iršič, Vesna (author), Rall, Douglas F. (author), Tuza, Zsolt (author)

Motivated by the success of domination games and by a variation of the coloring game called the indicated coloring game, we introduce a version of domination games called the indicated domination game. It is played on an arbitrary graph $G$ by two players, Dominator and Staller, where Dominator wants to finish the game in as few rounds as possible while Staller wants just the opposite. In each round, Dominator indicates a vertex $u$ of $G$ that has not been dominated by previous selections of Staller, which, by the rules of the game, forces Staller to select a vertex in the closed neighborhood of $u$. The game is finished when all vertices of $G$ become dominated by the vertices selected by Staller. Assuming that both players are playing optimally according to their goals, the number of selected vertices during the game is the indicated domination number, $\gamma_{\rm i}(G)$, of $G$. We prove several bounds on the indicated domination number expressed in terms of other graph invariants. In particular, we find a place of the new graph invariant in the well-known domination chain, by showing that $\gamma_{\rm i}(G)\ge \Gamma(G)$ for all graphs $G$, and by showing that the indicated domination number is incomparable with the game domination number and also with the upper irredundance number. In connection with the trivial upper bound $\gamma_{\rm i}(G)\le n(G)-\delta(G)$, we characterize the class of graphs $G$ attaining the bound provided that $n(G)\ge 2\delta(G)+2$. We prove that in trees, split graphs and grids the indicated domination number equals the independence number. We also find a formula for the indicated domination number of powers of paths, from which we derive that there exist graphs in which the indicated domination number is arbitrarily larger than the upper irredundance number. We provide some partial results supporting the statement that $\gamma_{\rm i}(G)=n(G)/2$ if $G$ is a cubic bipartite graph, and leave this as an open question.

• Language: English
• Keywords: domination game, indicated coloring, independence number, upper domination number
• Work type: Article
• Typology: 1.01 - Original Scientific Article
• Organization: FMF - Faculty of Mathematics and Physics
• Publication status: Published
• Publication version: Version of Record
• Year: 2024
• Number of pages: 10 str.
• Numbering: Vol. 347, iss. 9, art. 114060
• PID: 20.500.12556/RUL-156257
• UDC: 519.17
• ISSN on article: 0012-365X
• DOI: 10.1016/j.disc.2024.114060
• COBISS.SI-ID: 195595779
• Publication date in RUL: 16.05.2024

Record is a part of a journal

• Title: Discrete mathematics
• Shortened title: Discrete math.
• Publisher: Elsevier
• ISSN: 0012-365X
• COBISS.SI-ID: 1118479

Licences

• License: CC BY-NC 4.0, Creative Commons Attribution-NonCommercial 4.0 International
• Link: http://creativecommons.org/licenses/by-nc/4.0/
• Description: A creative commons license that bans commercial use, but the users don’t have to license their derivative works on the same terms.

Secondary language

• Language: Slovenian
• Keywords: dominacijska igra, indicirano barvanje, neodvisnostno število, zgornje dominantno število