The locating-dominating set problem is a special domination problem, challenging both from a theoretical and a computational point of view. Hence, a typical line of attack is to determine minimum locating-dominating sets of special graphs or to provide bounds for their size. Here, we study the locating-dominating set problem from a polyhedral point of view and demonstrate how the associated polyhedra can be entirely described for some basic families of graphs. The latter enables us to determine minimum weight locating-dominating sets in the studied graph classes for arbitrary integral node weights. We discuss further lines of research in order to apply similar techniques for other graph classes, to obtain the exact values or strong lower bounds for the size of minimum locating-dominating sets, stemming from linear relaxations of the polyhedra, enhanced by suitable cutting planes.