## Gravier, Sylvain and Janson, Svante and Laihonen, Tero and Ranto, Sanna - Graphs where every k-subset of vertices is an identifying set

dmtcs:1253 - Discrete Mathematics & Theoretical Computer Science, March 1, 2014, Vol. 16 no. 1
Graphs where every k-subset of vertices is an identifying set

Authors: Gravier, Sylvain and Janson, Svante and Laihonen, Tero and Ranto, Sanna

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.

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]