Characterizing extremal graphs for open neighbourhood location-domination

An open neighbourhood locating-dominating set is a S of vertices graph G such that each vertex has neighbour in S, and for any two u,v G, there at least one exactly u v. We characterize those graphs whose only the whole vertices. More precisely, we prove these are which all connected components half-graphs (a half-graph special bipartite with both parts same size, where part can be ordered so neighbourhoods consecutive differ by vertex). This corrects wrong characterization from literature.