Domination game and total domination game played on Sierpiński graphs

Abstract

The game domination numbers ${\gamma }_{\mathrm{g}}$ and ${\gamma }_{\mathrm{g}}^{\prime }$, and the game total domination numbers ${\gamma }_{\mathrm{tg}}$ and ${\gamma }_{\mathrm{tg}}^{\prime }$ are investigated on Sierpiński 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 $(2p-3)p^{n-2}\le {\gamma }_{\mathrm{g}}(S_p^n),{\gamma }_{\mathrm{g}}^{\prime }(S_p^n)\le (2p-2)p^{n-2}$. For the total domination game the bounds ${\gamma }_{\mathrm{tg}}(S_p^n),{\gamma }_{\mathrm{tg}}^{\prime }(S_p^n)\ge (2p-2)p^{n-2}$ are established.