On open neighborhood locating-dominating in graphs

Mustapha Chellali (LAMDA-Laboratory, Department of Mathematics, University of Blida, Blida, Algeria)
Nader Jafari Rad (Department of Mathematics, Shahrood University of Technology, Shahrood, Iran)
Suk Jai Seo (Computer Science Department, Middle Tennessee State University, Murfreesboro, TN 37132, U.S.A)
Peter James Slater (Mathematical Sciences Department and Computer Science Department, University of Alabama in Huntsville, Huntsville, AL 35899 USA)

Electronic Journal of Graph Theory and Applications (EJGTA)
Vol 2, No 2 (2014): Electronic Journal of Graph Theory and Applications

A set D of vertices in a graph G = (V (G), E(G)) is an open neighborhood locating-dominating set (OLD-set) for G if for every two vertices u, v of V (G) the sets N(u) ∩ D and N(v) ∩ D are non-empty and different. The open neighborhood locating-dominating number OLD(G) is the minimum cardinality of an OLD-set for G. In this paper we characterize graphs G of order n with OLD(G) = 2, 3, or n and graphs with minimum degree (G) ≥ 2 that are C4-free with OLD(G) = n-1.