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Graphs where every k-subset of vertices is an identifying set

Abstract : 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.
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Contributor : Sylvain Gravier <>
Submitted on : Wednesday, July 22, 2015 - 11:27:17 AM
Last modification on : Tuesday, October 6, 2020 - 8:36:02 AM
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  • HAL Id : hal-00362184, version 1
  • ARXIV : 0902.0443



Sylvain Gravier, Svante Janson, Tero Laihonen, Sanna Ranto. Graphs where every k-subset of vertices is an identifying set. Discrete Mathematics and Theoretical Computer Science, DMTCS, 2014, Vol. 16 no. 1 (in progress) (1), pp.73-88. ⟨hal-00362184⟩



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