Characterizing extremal digraphs for identifying codes and extremal cases of Bondy's theorem on induced subsets

Florent Foucaud (1) , Reza Naserasr (1, 2) , Aline Parreau (3)

1 LaBRI - Laboratoire Bordelais de Recherche en Informatique
2 LRI - Laboratoire de Recherche en Informatique
3 IF - Institut Fourier

An identifying code of a (di)graph $G$ is a dominating subset $C$ of the vertices of $G$ such that all distinct vertices of $G$ have distinct (in)neighbourhoods within $C$. In this paper, we classify all finite digraphs which only admit their whole vertex set in any identifying code. We also classify all such infinite oriented graphs. Furthermore, by relating this concept to a well known theorem of A.~Bondy on set systems we classify the extremal cases for this theorem.

Mots clés: Digraphs; Identifying codes; Dominating sets; Infinite digraphs; Induced sets

Domaines: Mathématique discrète [cs.DM]; Combinatoire [math.CO]

Florent Foucaud, Reza Naserasr, Aline Parreau. Characterizing extremal digraphs for identifying codes and extremal cases of Bondy's theorem on induced subsets. Graphs and Combinatorics, 2013, 29 (3), pp.463-473. ⟨10.1007/s00373-012-1136-4⟩. ⟨hal-00526446v3⟩