# Graphs where every k-subset of vertices is an identifying set - Mathematics > Combinatorics

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Abstract: Let $G=V,E$ be an undirected graph without loops and multiple edges. Asubset $C\subseteq V$ is called \emph{identifying} if for every vertex $x\in V$the intersection of $C$ and the closed neighbourhood of $x$ is nonempty, andthese 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 ofsuch a graph is at most $2k-2.$ Constructions attaining the maximal order aregiven for infinitely many values of $k.$The corresponding problem of $k$-subsets identifying any at most $\ell$vertices is considered as well.

Author: Sylvain Gravier, Svante Janson, Tero Laihonen, Sanna Ranto

Source: https://arxiv.org/