2010/04/30 by Florent Foucaud, Eleonora Guerrini, Matjaz Kovse +4 · 1 citation
Computer Science · Mathematics · #Advanced Graph Theory Research #Graph Labeling and Dimension Problems #Optimization and Search Problems #cs.DM #math.CO
paper · pdf · doi:10.1016/j.ejc.2011.01.002
published as European Journal of Combinatorics 32, 4 (2011) 628-638
openalex publication_date 2011/02/17 · arxiv created 2011/02/24 · arxiv updated 2011/02/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
An identifying code of a graph G is a dominating set C such that every vertex x of G is distinguished from all other vertices by the set of vertices in C that are at distance at most 1 from x. The problem of finding an identifying code of minimum possible size turned out to be a challenging problem. It was proved by N. Bertrand that if a graph on n vertices with at least one edge admits an identifying code, then a minimum identifying code has size at most n-1. Some classes of graphs whose smallest identifying code is of size n-1 were already known, and few conjectures were formulated to classify all these graphs. In this paper, disproving these conjectures, we classify all finite graphs for which all but one of the vertices are needed to form an identifying code. We also classify all infinite graphs needing the whole set of vertices in any identifying code. New upper bounds in terms of the number of vertices and the maximum degree of a graph are also provided.