Sylvain Gravier ; Svante Janson ; Tero Laihonen ; Sanna Ranto - 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: Sylvain Gravier ORCID-iD1,2; Svante Janson 3,4; Tero Laihonen ORCID-iD5; Sanna Ranto ORCID-iD6

  • 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.

Volume: Vol. 16 no. 1
Section: Combinatorics
Published on: March 1, 2014
Accepted on: July 23, 2015
Submitted on: February 16, 2010
Keywords: [MATH.MATH-CO] Mathematics [math]/Combinatorics [math.CO]

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Source : ScholeXplorer IsRelatedTo DOI 10.1016/j.ejc.2007.02.016
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