Given a finite graph G, the maximum length of a sequence \((v_1,\ldots ,v_k)\) of vertices in G such that each \(v_i\) dominates a vertex that is not dominated by any vertex in \(\{v_1,\ldots ,v_{i-1}\}\) is called the Grundy domination number, \(\gamma _\mathrm{gr}(G)\), 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 \(\gamma _\mathrm{gr}(G) \ge \frac{n + \lceil \frac{k}{2} \rceil - 2}{k-1}\) holds for every connected k-regular graph of order n different from \(K_{k+1}\) and \(\overline{2C_4}\). The bound in the case \(k=3\) reduces to \(\gamma _\mathrm{gr}(G)\ge \frac{n}{2}\), and we characterize the connected cubic graphs with \(\gamma _\mathrm{gr}(G)=\frac{n}{2}\). If G is different from \(K_4\) and \(K_{3,3}\), then \(\frac{n}{2}\) 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.

给定一个有限图ģ，一个序列的最大长度\（（V_1，\ ldots，V_K）\）在顶点ģ使得每个\（V_I \）占主导地位，其不以任何顶点为主顶点\（\ {V_1，\ ldots，V_ {I-1} \} \）被称为格伦迪控制数，\（\伽马_ \ mathrm {GR}（G）\） ，的ģ。对该定义进行少量修改即可产生Z-格伦迪控制数，它是众所周知的零强迫数的对偶不变式。在本文中，我们证明\（\ gamma _ \ mathrm {gr}（G）\ ge \ frac {n + \ lceil \ frac {k} {2} \ rceil-2} {k-1} \成立对于阶n的每个连接的k-正则图与\（K_ {k + 1} \）和\（\ overline {2C_4} \）不同。在\（k = 3 \）的情况下边界变为\（\ gamma _ \ mathrm {gr}（G）\ ge \ frac {n} {2} \），我们用\（ \ gamma _ \ mathrm {gr}（G）= \ frac {n} {2} \）。如果G与\（K_4 \）和\（K_ {3,3} \）不同，则\（\ frac {n} {2} \）也是连通三次图的零逼近数的上限，并刻画达到此界限的连通三次图。