Minimum sizes of identifying codes in graphs differing by one edge

  • Authors:

  • Irène Charon;Iiro Honkala;Olivier Hudry;Antoine Lobstein

  • Affiliations:

  • Institut Télécom - Télécom ParisTech & CNRS - LTCI UMR 5141, Paris Cedex 13, France 75634;Department of Mathematics and Statistics, University of Turku, Turku, Finland 20014;Institut Télécom - Télécom ParisTech & CNRS - LTCI UMR 5141, Paris Cedex 13, France 75634;Institut Télécom - Télécom ParisTech & CNRS - LTCI UMR 5141, Paris Cedex 13, France 75634

  • Venue:
  • Cryptography and Communications
  • Year:
  • 2014

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Abstract

Let G be a simple, undirected graph with vertex set V. For v 驴 V and r 驴 1, we denote by B G, r (v) the ball of radius r and centre v. A set 𝒞 ⊆ V ${\mathcal C} \subseteq V$ is said to be an r-identifying code in G if the sets B G , r ( v ) 驴 𝒞 $B_{G,r}(v)\cap {\mathcal C}$ , v 驴 V, are all nonempty and distinct. A graph G admitting an r-identifying code is called r-twin-free, and in this case the size of a smallest r-identifying code in G is denoted by 驴 r (G). We study the following structural problem: let G be an r-twin-free graph, and G 驴 be a graph obtained from G by adding or deleting an edge. If G 驴 is still r-twin-free, we compare the behaviours of 驴 r (G) and 驴 r (G 驴), establishing results on their possible differences and ratios.