Locating-dominating sets: From graphs to oriented graphs

A locating-dominating set of an undirected graph is a subset vertices S such that dominating and for every u,v?S, the neighbourhood u v on are distinct (i.e. N(u)?S?N(v)?S). Locating-dominating sets have received considerable attention in last decades. In this paper, we consider oriented version problem. each w?V?S, N?(w)?S?? pair u,v?V?S, N?(u)?S?N?(v)?S. We following two parameters. Given G, look ??LD(G) (??LD(G)) which size smallest (largest) optimal over all orientations G. particular, if D orientation then ??LD(G)??LD(D)???LD(G) where ?LD(D) minimum D. For best orientation, prove that, twin-free G n vertices, ??LD(G)?n/2 proves “directed version” widely studied conjecture location-domination number. As side result obtain new improved upper bound number trees. Moreover, give some bounds many classes drastically improve value n/2 (almost) d-regular graphs by showing ??LD(G)?O(log?d/d?n) using probabilistic argument. While ??LD(G)??LD(G) holds as outerplanar ??LD(G)??LD(G) ??LD(G)??LD(G) complete graphs. also general ??LD(G) ??LD(G)??(G). Finally, show polynomial but leave open question whether there exist with ??LD(G)?O(log?n).