Resolving sets for Johnson and Kneser graphs

TL;DR

This paper explores methods to construct resolving sets in Johnson and Kneser graphs, utilizing combinatorial objects like projective planes, designs, and matrices to improve understanding of these graph classes.

Contribution

It introduces new constructions of resolving sets for Johnson and Kneser graphs using diverse combinatorial structures, expanding the toolkit for graph resolving set analysis.

Findings

Various combinatorial objects can be used to construct resolving sets.

New explicit constructions for Johnson and Kneser graphs.

Connections between combinatorial designs and graph resolving sets.

Abstract

A set of vertices $S$ in a graph $G$ is a {\em resolving set} for $G$ if, for any two vertices $u,v$, there exists $x\in S$ such that the distances $d(u,x) \neq d(v,x)$. In this paper, we consider the Johnson graphs $J(n,k)$ and Kneser graphs $K(n,k)$, and obtain various constructions of resolving sets for these graphs. As well as general constructions, we show that various interesting combinatorial objects can be used to obtain resolving sets in these graphs, including (for Johnson graphs) projective planes and symmetric designs, as well as (for Kneser graphs) partial geometries, Hadamard matrices, Steiner systems and toroidal grids.