On trees with unique locating kernels

Abstract

AbstractA locating-dominating set of a graph G is a set D of vertices such that for every two vertices $$x,y\in V(G)\setminus D$$ x , y ∈ V ( G ) \ D the sets $$N(x)\cap D$$ N ( x ) ∩ D and $$N(y)\cap D$$ N ( y ) ∩ D 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.