On the length of L-Grundy sequences

An L- sequence of a graph $G$ is a sequence of distinct vertices $S = \{v_1, ... , v_k\}$ such that $N[v_i] \setminus \cup_{j=1}^{i-1} N(v_j) \neq \emptyset$. The length of the longest L-sequence is called the L-Grundy domination number, denoted $\gamma_{gr}^L(G)$. In this paper, we prove $\gamma_{gr}^L(G) \leq n(G) - \delta(G) + 1$, which was conjectured by Bre{\v{s}}ar, Gologranc, Henning, and Kos. We also prove some early results about characteristics of $n$-vertex graphs such $\gamma_{gr}^L(G) = n$, as well as bounds on the change in L-Grundy number under graph operations.