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

Combinatorics 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...

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Bibliographic Details
Published in:Discrete mathematics and theoretical computer science 2014-01, Vol.16 no. 1 (Combinatorics), p.73-88
Main Authors: Gravier, Sylvain, Janson, Svante, Laihonen, Tero, Ranto, Sanna
Format: Article
Language:English
Subjects:
[math.math-co] mathematics [math]/combinatorics [math.co]
Combinatorics
extremal graph
Graphs
identifying code
Mathematics
Plotkin bound
strongly regular graph
Theorems
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Summary:Combinatorics 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.
ISSN:1365-8050
1462-7264
1365-8050
DOI:10.46298/dmtcs.1253