Maker-Breaker domination game on trees when Staller wins

Abstract

In the Maker-Breaker domination game played on a graph $G$, Dominator's goal is to select a dominating set and Staller's goal is to claim a closed neighborhood of some vertex. We study the cases when Staller can win the game. If Dominator (resp., Staller) starts the game, then $γ_{\rm SMB}(G)$ (resp., $γ_{\rm SMB}'(G)$) denotes the minimum number of moves Staller needs to win. For every positive integer $k$, trees $T$ with $γ_{\rm SMB}'(T)=k$ are characterized and a general upper bound on $γ_{\rm SMB}'$ is proved. Let $S = S(n_1,\dots, n_\ell)$ be the subdivided star obtained from the star with $\ell$ edges by subdividing its edges $n_1-1, \ldots, n_\ell-1$ times, respectively. Then $γ_{\rm SMB}'(S)$ is determined in all the cases except when $\ell\ge 4$ and each $n_i$ is even. The simplest formula is obtained when there are at least two odd $n_i$s. If $n_1$ and $n_2$ are the two smallest such numbers, then $γ_{\rm SMB}'(S(n_1,\dots, n_\ell))=\lceil \log_2(n_1+n_2+1)\rceil$. For caterpillars, exact formulas for $γ_{\rm SMB}$ and for $γ_{\rm SMB}'$ are established.

Figures (2)

• Figure 1: Families ${\cal S}_1$, ${\cal S}_2$, ${\cal S}_3$, and ${\cal S}_4$, where the black vertices are the vertices with fixed degrees.
• Figure 2: Caterpillar $T$.

Theorems & Definitions (31)

• Proposition 2.1: bujtas_fast_2022
• Proposition 2.2: bujtas_fast_2022
• Proposition 2.3: bujtas_fast_2022
• Proposition 2.4: bujtas_fast_2022
• Proposition 2.5: bujtas_fast_2022
• Theorem 2.6: duchene_makerbreaker_2020gledel_makerbreaker_2019
• Lemma 2.7: bujtas_fast_2022
• Theorem 2.8: bujtas_fast_2022
• Lemma 3.1
• proof