Trees with Maximum p-Reinforcement Number

Let $G=(V,E)$ be a graph and $p$ a positive integer. The $p$-domination number ${\text{g}}_p(G)$ is the minimum cardinality of a set $D\subseteq V$ with $\mathrm{\mid }{N}_G(x)\cap D\mathrm{\mid }\ge p$ 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 $G^{\mathrm{\prime }}$ with ${\text{g}}_p(G^{\mathrm{\prime }})<{\text{g}}_p(G)$. Recently, it was proved by Lu et al. that $r_p(T)\le p+1$ for a tree $T$ and $p\ge 2$. In this paper, we characterize all trees attaining this upper bound for $p\ge 3$.