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, gamma(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(gr)(G) >= n+inverted right perpendiculark/2inverted left perpendicular-2/k-1 holds for every connected k-regular graph of order n different from Kk+1 and (2C(4)) over bar. The bound in the case k = 3 reduces to gamma(gr)(G) >= n/2, and we characterize the connected cubic graphs with gamma(gr)(G) = n/2. If G is different from K-4 and K-3,K-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.