Kirszbraun-type Theorems For Graphs

TL;DR

This paper extends Kirszbraun's theorem to graph metric spaces, characterizing when graphs allow Lipschitz extensions and exploring the associated Helly property and computational complexity.

Contribution

It characterizes $bZ^d$-Kirszbraun graphs via a Helly property and analyzes the complexity of these extension properties in graph metrics.

Findings

$bZ^d$-Kirszbraun graphs are exactly those satisfying a specific Helly property.

The paper establishes the relationship between Lipschitz extension properties and Helly-type conditions in graphs.

Complexity results for deciding the Kirszbraun property in graph metric spaces.

Abstract

The classical Kirszbraun theorem says that all $1$-Lipschitz functions $f:A\longrightarrow \mathbb{R}^n$, $A\subset \mathbb{R}^n$, with the Euclidean metric have a $1$-Lipschitz extension to $\mathbb{R}^n$. For metric spaces $X,Y$ we say that $Y$ is $X$-Kirszbraun if all $1$-Lipschitz functions $f:A\longrightarrow Y$, $A\subset X$, have a $1$-Lipschitz extension to~$X$. We analyze the case when $X$ and $Y$ are graphs with the usual path metric. We prove that $\mathbb{Z}^d$-Kirszbraun graphs are exactly graphs that satisfies a certain Helly property. We also consider complexity aspects of these properties.