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Paper   IPM / M / 16723
School of Mathematics
  Title:   Characterizing extremal graphs for open neighbourhood location-domination
1.  Narges Ghareghani
2.  P Sharifani (Joint with F. Foucaud and A. Roshany-Tabrizi)
  Status:   Published
  Journal: Discrete Appl. Math.
  Vol.:  302
  Year:  2021
  Pages:   76-79
  Supported by:  IPM
An open neighbourhood locating-dominating set is a set S of vertices of a graph G such that each vertex of G has a neighbour in S, and for any two vertices u, v of G, there is at least one vertex in S that is a neighbour of exactly one of u and v. We characterize those graphs whose only open neighbourhood locating-dominating set is the whole set of vertices. More precisely, we prove that these graphs are exactly the graphs for which all connected components are half-graphs (a half-graph is a special bipartite graph with both parts of the same size, where each part can be ordered so that the open neighbourhoods of consecutive vertices differ by exactly one vertex). This corrects a wrong characterization from the literature.

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