On the length of L-Grundy sequences

Abstract

An L- sequence of a graph $G$ is a sequence of distinct vertices $S=\{v_1,...,v_k\}$ such that $N[v_i]\setminus {\cup }_{j=1}^{i-1}N(v_j)\ne \varnothing$. The length of the longest L-sequence is called the L-Grundy domination number, denoted ${\gamma }_{gr}^L(G)$. In this paper, we prove ${\gamma }_{gr}^L(G)\le n(G)-\delta (G)+1$, which was conjectured by Bre{š}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.