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.

Source : oai:HAL:hal-00362184v1

Volume: Vol. 16 no. 1

Section: Combinatorics

Published on: March 1, 2014

Submitted on: February 16, 2010

Keywords: [MATH.MATH-CO] Mathematics [math]/Combinatorics [math.CO]

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