Grundy Domination and Zero Forcing in Regular Graphs

Given a finite graph G , the maximum length of a sequence (v_1,… ,v_k) of vertices in G such that each v_i dominates a vertex that is not dominated by any vertex in {v_1,… ,v_i-1} is called the Grundy domination number, γ _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 γ _gr(G) ≥n + ⌈k/2⌉ - 2/k-1 holds for every connected k -regular graph of order n different from K_k+1 and 2C_4 . The bound in the case k=3 reduces to γ _gr(G)≥n/2 , and we characterize the connected cubic graphs with γ _gr(G)=n/2 . If G is different from K_4 and K_3,3 , then 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.