Open-independent, open-locating-dominating sets: structural aspects of some classes of graphs

Let G = (V (G), E(G)) be a finite simple undirected graph with vertex set V (G), edge set E(G) and vertex subset S subset of V (G). S is termed open-dominating if every vertex of G has at least one neighbor in S, and open-independent, open-locating-dominating (an OLDoind-set for short) if no two vertices in G have the same set of neighbors in S, and each vertex in S is open-dominated exactly once by S. The problem of deciding whether or not G has an OLDoind-set has important applications that have been reported elsewhere. As the problem is known to be NP-complete, it appears to be notoriously difficult as we show that its complexity remains the same even for just planar bipartite graphs of maximum degree five and girth six, and also for planar subcubic graphs of girth nine. Also, we present characterizations of both P-4-tidy graphs and the complementary prisms of cographs that have an OLDoind-set.