Grundy Domination and Zero Forcing in Regular Graphs

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