The connectivity dimension of a graph

TL;DR

This paper introduces the connectivity dimension of a graph, a new measure of connectivity heterogeneity, characterizes extremal cases, provides constructions, links it to classical graph concepts, and proves its computation is NP-complete.

Contribution

It defines and explores the connectivity dimension, establishing bounds, constructions, and complexity results, thus expanding graph theory parameters.

Findings

Characterized extremal graphs for connectivity dimension

Constructed graphs with prescribed connectivity dimension

Proved computing the connectivity dimension is NP-complete

Abstract

This article investigates the connectivity dimension of a graph. We introduce this concept in analogy to the metric dimension of a graph, providing a graph parameter that measures the heterogeneity of the connectivity structure of a graph. We fully characterize extremal examples and present explicit constructions of infinitely many graphs realizing any prescribed non-extremal connectivity dimension. We also establish a general lower bound in terms of the graph's block structure, linking the parameter to classical notions from graph theory. Finally, we prove that the problem of computing the connectivity dimension is NP-complete.