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

TL;DR

This paper investigates graphs where every k-subset of vertices uniquely identifies all vertices, establishing an upper bound on their size and providing constructions for infinitely many k values.

Contribution

It proves an upper bound on the size of graphs where every k-subset is identifying and offers constructions achieving this bound for infinitely many k.

Findings

Maximum order of such graphs is at most 2k-2 for each k>1.

Constructions exist for infinitely many k values that attain this maximum.

The problem extends to k-subsets identifying up to  vertices.

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.