Extremal cardinalities for identifying and locating-dominating codes in graphs

Abstract

Consider a connected undirected graph G = (V ,E), a subset of vertices C ⊆ V , and an integer r 1; for any vertex v ∈ V , let Br(v) denote the ball of radius r centred at v, i.e., the set of all vertices linked to v by a path of at most r edges. If for all vertices v ∈ V (respectively, v ∈ V \C), the setsBr(v)∩C are all nonempty and different, then we call C an r-identifying code (respectively, an r-locating-dominating code). We study the extremal values of the cardinality of a minimum r-identifying or r-locating-dominating code in any connected undirected graphG having a given number, n, of vertices. It is known that aminimum r-identifying code contains at least log2(n+1) vertices; we establish in particular that such a code contains at most n− 1 vertices, and we prove that these two bounds are reached. The same type of results are given for locating-dominating codes. © 2006 Elsevier B.V. All rights reserved.

DOI: 10.1016/j.disc.2005.09.027

Cite this paper

@article{Charon2007ExtremalCF, title={Extremal cardinalities for identifying and locating-dominating codes in graphs}, author={Ir{\`e}ne Charon and Olivier Hudry and Antoine Lobstein}, journal={Discrete Mathematics}, year={2007}, volume={307}, pages={356-366} }