Open-independent, open-locating-dominating sets: structural aspects of some classes of graphs
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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 OLD_oind-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 OLD_oind-set has important applications, has been studied elsewhere and is known to be 𝒩𝒫-complete. Reinforcing this result, it appears that the problem is notoriously difficult as we show that its complexity remains the same even for just planar bipartite graphs. Also, we present characterizations of both P_4-tidy graphs and the complementary prisms of cographs, that have an OLD_oind-set.
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