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 ⊆ 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 and also for planar subcubic graphs. Also, we present characterizations of both P4-tidy graphs and the complementary prisms of cographs that have an OLDoind-set.

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