A1 Refereed original research article in a scientific journal
On r-locating-dominating sets in paths
Authors: Honkala I
Publisher: ACADEMIC PRESS LTD- ELSEVIER SCIENCE LTD
Publication year: 2009
Journal: European Journal of Combinatorics
Journal name in source: EUROPEAN JOURNAL OF COMBINATORICS
Journal acronym: EUR J COMBIN
Volume: 30
Issue: 4
First page : 1022
Last page: 1025
Number of pages: 4
ISSN: 0195-6698
DOI: https://doi.org/10.1016/j.ejc.2008.04.011(external)
Abstract
Assume that G = (V, E) is a simple undirected graph, and C is a nonempty subset of V. For every v is an element of V, we define I(r)(v) = {u is an element of C | d(G)(u, v) <= r}, where d(G)(u, v) denotes the number of edges on any shortest path between u and v. If the sets I(r)(v) for v is not an element of C are pairwise different, and none of them is the empty set, we say that C is an r-locating-dominating set in G. It is shown that the smallest 2-locating-dominating set in a path with n vertices has cardinality inverted right perpendicular (n + 1)/3 inverted left perpendicular, which coincides with the lower bound proved earlier by Bertrand, Charon, Hudry and Lobstein. Moreover, we give a general upper bound which improves a result of Bertrand, Charon, Hudry and Lobstein. (C) 2008 Elsevier Ltd. All rights reserved.
Assume that G = (V, E) is a simple undirected graph, and C is a nonempty subset of V. For every v is an element of V, we define I(r)(v) = {u is an element of C | d(G)(u, v) <= r}, where d(G)(u, v) denotes the number of edges on any shortest path between u and v. If the sets I(r)(v) for v is not an element of C are pairwise different, and none of them is the empty set, we say that C is an r-locating-dominating set in G. It is shown that the smallest 2-locating-dominating set in a path with n vertices has cardinality inverted right perpendicular (n + 1)/3 inverted left perpendicular, which coincides with the lower bound proved earlier by Bertrand, Charon, Hudry and Lobstein. Moreover, we give a general upper bound which improves a result of Bertrand, Charon, Hudry and Lobstein. (C) 2008 Elsevier Ltd. All rights reserved.
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