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On open neighborhood locating-dominating in graphs. (English) Zbl 1306.05178

Summary: 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 \(N(u)\cap D\) and \(N(v)\cap D\) are non-empty and different. The open neighborhood locating-dominating number \(\operatorname{OLD}(G)\) is the minimum cardinality of an OLD-set for \(G\). In this paper, we characterize graphs \(G\) of order \(n\) with \(\operatorname{OLD}(G)=2\), 3, or \(n\) and graphs with minimum degree \(\delta(G)\geq 2\) that are \(C_4\)-free with \(\operatorname{OLD}(G)= n-1\).

MSC:

05C69 Vertex subsets with special properties (dominating sets, independent sets, cliques, etc.)
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