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  • On the game domination number of graphs with given minimum degree [Elektronski vir]
    Bujtás, Csilla
    In the domination game, introduced by B. Brešar et al. [SIAM J. Discrete Math. 24, No. 3, 979-991 (2010)], Dominator and Staller alternately select a vertex of a graph ▫$G$▫. A move is legal if the ... selected vertex ▫$v$▫ dominates at least one new vertex - that is, if we have a ▫$u\in N[v]$▫ for which no vertex from ▫$N[u]$▫ was chosen up to this point of the game. The game ends when no more legal moves can be made, and its length equals the number of vertices selected. The goal of Dominator is to minimize whilst that of Staller is to maximize the length of the game. The game domination number ▫$\gamma_g(G)$▫ of ▫$G$▫ is the length of the domination game in which Dominator starts and both players play optimally. In this paper we establish an upper bound on ▫$\gamma_g(G)$▫ in terms of the minimum degree ▫$\delta$▫ and the order ▫$n$▫ of ▫$G$▫. Our main result states that for every ▫$\delta \geq 4$▫, ▫$$ \gamma_g(G)\leq \frac{15\delta^4-28\delta^3-129\delta^2+354\delta-216}{45\delta^4-195\delta^3+174\delta^2+174\delta-216}\; n. $$▫ Particularly, ▫$\gamma_g(G) < 0.5139\; n$▫ holds for every graph of minimum degree 4, and ▫$\gamma_g(G) < 0.4803\; n$▫ if the minimum degree is greater than 4. Additionally, we prove that ▫$\gamma_g(G) < 0.5574\; n$ if $\delta=3$▫.
    Vir: The Electronic journal of combinatorics [Elektronski vir]. - ISSN 1077-8926 (Vol. 22, iss. 3, 2015, art. P3.29 (18 str.))
    Vrsta gradiva - e-članek
    Leto - 2015
    Jezik - angleški
    COBISS.SI-ID - 18893145

    Povezava(-e):

    https://www.combinatorics.org/ojs/index.php/eljc/article/view/v22i3p29/pdf
    DOI

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