The p-Bondage Number of Trees

Let p be a positive integer and G = (V, E) be a simple graph. A p-dominating set of G is a subset \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${D\,{\subseteq}\, V}$$\end{document} such that every vertex not in D has at least p neighbors in D. The p-domination number of G is the minimum cardinality of a p-dominating set of G. The p-bondage number of a graph G with (ΔG) ≥ p is the minimum cardinality among all sets of edges \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${B\subseteq E}$$\end{document} for which γp(G − B) > γp(G). For any integer p ≥ 2 and tree T with (ΔT) ≥ p, this paper shows that 1 ≤  bp(T) ≤ (ΔT) − p + 1, and characterizes all trees achieving the equalities.