Daisy Hamming graphs

TL;DR

This paper characterizes when daisy graphs of rooted graphs, especially Hamming graphs, are isometric, providing solutions to open problems and new characterizations of these subgraphs.

Contribution

It solves an open problem by characterizing rooted graphs with isometric daisy graphs and provides new characterizations of isometric daisy graphs in Hamming graphs.

Findings

Characterized rooted graphs with all daisy graphs isometric in $G$.

Identified conditions for daisy graphs of size 2 to be isometric in Hamming graphs.

Provided a new characterization of isometric daisy graphs in Cartesian products of complete graphs.

Abstract

Daisy graphs of a rooted graph $G$ with the root $r$ were recently introduced as a generalization of daisy cubes, a class of isometric subgraphs of hypercubes. In this paper we first solve the problem posed in \cite{Taranenko2020} and characterize rooted graphs $G$ with the root $r$ for which all daisy graphs of $G$ with respect to $r$ are isometric in $G$. We continue the investigation of daisy graphs $G$ (generated by $X$) of a Hamming graph $H$ and characterize those daisy graphs generated by $X$ of cardinality 2 that are isometric in $H$. Finally, we give a characterization of isometric daisy graphs of a Hamming graph $K_{k_1}\Box \ldots \Box K_{k_n}$ with respect to $0^n$ in terms of an expansion procedure.