Well-located graphs: a collection of well-covered ones Original Research Article

A well-covered graph is one in which every maximal independent set is 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 (Congr. Numer. 65 (1988) 191–200) the authors showed that the well-located graphs are a subclass of the well-covered ones and, for girth 5, the two classes are identical. In this article, we begin the examination of the well-located graphs of girth 4 and, for those well-located graphs with no 4-cycles (but 3-cycles are allowed), obtain a characterization.