On the length of L-Grundy sequences

Abstract

An L-sequence of a graph $G$ is a sequence of distinct vertices $S=(v_1,\dots ,v_k)$ such that $N\left[v_i\right]\setminus {\cup }_{j=1}^{i-1}N\left(v_j\right)\ne 0̸$. The length of a longest L-sequence is called the L-Grundy domination number, denoted ${\gamma }_{gr}^L\left(G\right)$. In this paper, we prove ${\gamma }_{gr}^L\left(G\right)\le n\left(G\right)-\delta \left(G\right)+1$, which was conjectured by Brešar, Gologranc, Henning, and Kos. We also prove some initial results about characteristics of $n$-vertex graphs satisfying ${\gamma }_{gr}^L\left(G\right)=n$.