On trees with unique locating kernels

A locating-dominating set of a graph G is a set D of vertices such that for every two vertices x,y∈V(G)\D\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x,y\in V(G)\setminus D$$\end{document} the sets N(x)∩D\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N(x)\cap D$$\end{document} and N(y)∩D\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N(y)\cap D$$\end{document} are non-empty and different. In this paper, we define the locating kernel of a graph G, i.e., a subset of its vertex set which is independent and a locating-dominating set. We provide a constructive characterization of trees with a unique locating kernel.