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On Connected 3-\(\gamma _{L}\)-Dot-Critical Graphs

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Abstract

A set D of vertices in a graph G is a locating-dominating set if for every two vertices uv of \(V-D\) the sets \(N_{G}(u)\cap D\) and \(N_{G}(v)\cap D\) are non-empty and different. The locating-domination number \(\gamma _{L}(G)\) is the minimum cardinality of a locating-dominating set of G. A graph G is \(\gamma _{L}\)-dot-critical if contracting any edge of G decreases its locating-domination number. Let \(k\ge 3\) be an integer. A graph G is called k-\(\gamma _{L}\)-dot-critical if G is \(\gamma _{L}\)-dot-critical with \(\gamma _{L}\left( G\right) =k.\) In this paper, we characterize all connected 3-\(\gamma _{L}\)-dot-critical graphs.

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Correspondence to Noureddine Ikhlef-Eschouf.

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Communicated by Ebrahim Ghorbani.

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Mimouni, M., Ikhlef-Eschouf, N. & Zamime, M. On Connected 3-\(\gamma _{L}\)-Dot-Critical Graphs. Bull. Iran. Math. Soc. 48, 979–991 (2022). https://doi.org/10.1007/s41980-021-00558-y

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  • DOI: https://doi.org/10.1007/s41980-021-00558-y

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