On r-locating-dominating sets in paths



: Honkala I

Publisher: ACADEMIC PRESS LTD- ELSEVIER SCIENCE LTD

: 2009

: European Journal of Combinatorics

: EUROPEAN JOURNAL OF COMBINATORICS

: EUR J COMBIN

: 30

: 4

: 1022

: 1025

: 4

: 0195-6698

DOI: https://doi.org/10.1016/j.ejc.2008.04.011


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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