On the length of L-Grundy sequences

TL;DR

This paper investigates the L-Grundy domination number in graphs, proving an upper bound related to graph order and minimum degree, and explores properties and bounds of this parameter under various graph operations.

Contribution

It establishes a proven upper bound for the L-Grundy domination number, confirming a conjecture, and analyzes its behavior under graph modifications.

Findings

Proved the bound $oxed{ ext{γ}_{ ext{gr}}^L(G) extless= n(G) - ext{δ}(G) + 1}$.

Characterized graphs with maximum L-Grundy number equal to the number of vertices.

Derived bounds on how the L-Grundy number changes with graph operations.

Abstract

An L- sequence of a graph $G $ is a sequence of distinct vertices $S = \{v_1, ... , v_k\}$ such that $N[v_i] \setminus \cup_{j=1}^{i-1} N(v_j) \neq \emptyset$. 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.