Moving closer: contractive maps on discrete metric spaces and graphs

TL;DR

This paper investigates the properties of contractive maps on discrete metric spaces and graphs, characterizing spaces with non-constant contractions and exploring their relationships with other functions.

Contribution

It introduces the concept of contractive maps on discrete spaces and characterizes those spaces that admit non-constant contractions, extending the theory to graph substructures.

Findings

Characterization of spaces with non-constant contractive maps

Examples illustrating relations between contractions and constant functions

Application of results to subgraphs of connected graphs

Abstract

We consider discrete metric spaces and we look for non-constant contractions. We introduce the notion of contractive map and we characterize the spaces with non-constant contractive maps. We provide some examples to discussion the possible relations between contractions, contractive maps and constant functions. Finally we apply the main result to the subgraphs of a non-oriented, connected graph.