Trees with Maximum p-Reinforcement Number

TL;DR

This paper characterizes all trees that reach the maximum p-reinforcement number of p+1 for p≥3, expanding understanding of how edge additions influence p-domination in trees.

Contribution

It provides a complete characterization of trees with maximum p-reinforcement number, specifically those attaining the upper bound of p+1 for p≥3.

Findings

Identifies all trees with r_p(T) = p+1 for p≥3.

Extends previous bounds on p-reinforcement numbers in trees.

Provides structural insights into trees with maximum reinforcement number.

Abstract

Let $G=(V,E)$ be a graph and $p$ a positive integer. The $p$-domination number $\g_p(G)$ is the minimum cardinality of a set $D\subseteq V$ with $|N_G(x)\cap D|\geq p$ for all $x\in V\setminus D$. The $p$-reinforcement number $r_p(G)$ is the smallest number of edges whose addition to $G$ results in a graph $G'$ with $\g_p(G')<\g_p(G)$. Recently, it was proved by Lu et al. that $r_p(T)\leq p+1$ for a tree $T$ and $p\geq 2$. In this paper, we characterize all trees attaining this upper bound for $p\geq 3$.