Well-located graphs: A collection of well-covered ones

A well-covered graph is one in which every maximal independent set is a maximum. A dominating set, say D, is said to be locating if for every pair of vertices not in the set D their neighbours that are in D differ in at least one vertex. A graph is called well-located if it has the property that every independent dominating set is locating. In a previous paper the authors showed that the well-located graphs are, in fact, well-covered and, for girth 5, are precisely the well-covered collection. In this article we begin the examination of the well-located graphs of girth 4 as well as those with no 4-cycles (but 3- cycles are allowed).