On locating-domination in graphs

A set D of vertices in a graph G = (V,E) is a locating-dominating set (LDS) if for every two vertices u,v of V-D the sets N(u)∩ D and N(v)∩ D are non-empty and different. The locating-domination number ${\displaystyle {\gamma }_L\left(G\right)}$ is the minimum cardinality of a LDS of G, and the upper locating-domination number, ${\displaystyle {\Gamma }_L\left(G\right)}$ is the maximum cardinality of a minimal LDS of G. We present different bounds on ${\displaystyle {\Gamma }_L\left(G\right)}$ and ${\displaystyle {\gamma }_L\left(G\right)}$.