Authors: Sylvain Gravier ^1,2; Svante Janson ^3,4; Tero Laihonen ⁵; Sanna Ranto ⁶

• 1 Institut Fourier
• 2 Laboratoire Leibniz
• 3 Department of Mathematics [Uppsala]
• 4 Uppsala University
• 5 Coding Theory Research Group
• 6 Department of Physics and Astronomy [Turku]

Let $G=(V,E)$ be an undirected graph without loops and multiple edges. A subset $C\subseteq V$ is called \emph{identifying} if for every vertex $x\in V$ the intersection of $C$ and the closed neighbourhood of $x$ is nonempty, and these intersections are different for different vertices $x$. Let $k$ be a positive integer. We will consider graphs where \emph{every} $k$-subset is identifying. We prove that for every $k>1$ the maximal order of such a graph is at most $2k-2.$ Constructions attaining the maximal order are given for infinitely many values of $k.$ The corresponding problem of $k$-subsets identifying any at most $\ell$ vertices is considered as well.