Given a finite graph G, the maximum length of a sequence (v1,…,vk)\documentclass[12pt]{minimal}
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\begin{document}$$(v_1,\ldots ,v_k)$$\end{document} of vertices in G such that each vi\documentclass[12pt]{minimal}
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\begin{document}$$v_i$$\end{document} dominates a vertex that is not dominated by any vertex in {v1,…,vi-1}\documentclass[12pt]{minimal}
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\begin{document}$$\{v_1,\ldots ,v_{i-1}\}$$\end{document} is called the Grundy domination number, γgr(G)\documentclass[12pt]{minimal}
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\begin{document}$$\gamma _\mathrm{gr}(G)$$\end{document}, of G. A small modification of the definition yields the Z-Grundy domination number, which is the dual invariant of the well-known zero forcing number. In this paper, we prove that γgr(G)≥n+⌈k2⌉-2k-1\documentclass[12pt]{minimal}
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\begin{document}$$\gamma _\mathrm{gr}(G) \ge \frac{n + \lceil \frac{k}{2} \rceil - 2}{k-1}$$\end{document} holds for every connected k-regular graph of order n different from Kk+1\documentclass[12pt]{minimal}
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\begin{document}$$K_{k+1}$$\end{document} and 2C4¯\documentclass[12pt]{minimal}
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\begin{document}$$\overline{2C_4}$$\end{document}. The bound in the case k=3\documentclass[12pt]{minimal}
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\begin{document}$$k=3$$\end{document} reduces to γgr(G)≥n2\documentclass[12pt]{minimal}
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\begin{document}$$\gamma _\mathrm{gr}(G)\ge \frac{n}{2}$$\end{document}, and we characterize the connected cubic graphs with γgr(G)=n2\documentclass[12pt]{minimal}
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\begin{document}$$\gamma _\mathrm{gr}(G)=\frac{n}{2}$$\end{document}. If G is different from K4\documentclass[12pt]{minimal}
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\begin{document}$$K_4$$\end{document} and K3,3\documentclass[12pt]{minimal}
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\begin{document}$$K_{3,3}$$\end{document}, then n2\documentclass[12pt]{minimal}
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\begin{document}$$\frac{n}{2}$$\end{document} is also an upper bound for the zero forcing number of a connected cubic graph, and we characterize the connected cubic graphs attaining this bound.
