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arXiv · 2608.21225

Domination game and total domination game played on Sierpi\'{n}ski graphs

Tanja Dravec·Daniel P. Johnston·Sandi Klavžar
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Abstract

The game domination numbers $\gamma_{\rm g}$ and $\gamma_{\rm g}^\prime$, and the game total domination numbers $\gamma_{{\rm{tg}}}$ and $\gamma_{{\rm{tg}}}^\prime$ are investigated on Sierpi\'{n}ski graphs $S_p^n$. For the domination game, the trivial bounds arising from the known domination number and the Grundy domination number of $S_p^n$ are significantly improved by proving that $(2p-3)p^{n-2}\leq \gamma_{\rm g}(S_p^n), \gamma_{\rm g}^\prime(S_p^n) \leq (2p-2)p^{n-2}$. For the total domination game the bounds $\gamma_{{\rm{tg}}}(S_p^n), \gamma_{{\rm{tg}}}^\prime(S_p^n) \geq (2p-2)p^{n-2}$ are established.

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Tanja Dravec, Daniel P. Johnston, Sandi Klavžar. 2026-08-21. Domination game and total domination game played on Sierpi\'{n}ski graphs. https://arxiv.org/abs/2608.21225

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