Graphs that are minor minimal with respect to dimension

TL;DR

This paper explores the concept of graph dimension as defined by Erdős, Harary, and Tutte, focusing on computing exact dimensions and identifying graphs that are minor minimal with respect to their dimension.

Contribution

It introduces conditions for easier computation of graph sums' dimensions and constructs three infinite classes of graphs that are minor minimal regarding their dimension.

Findings

Established conditions for computing dimensions of graph sums

Constructed three infinite classes of minor minimal graphs

Advanced understanding of graph dimension and minimality

Abstract

Erd\H{o}s, Harary, and Tutte defined the dimension of a graph $G$ as the smallest natural number $n$ such that $G$ can be embedded in $\mathbb{R}^n$ with each edge a straight line segment of length 1. Since the proposal of this definition, little has been published on how to compute the exact dimension of graphs and almost nothing has been published on graphs that are minor minimal with respect to dimension. This paper develops both of these areas. In particular, it (1) establishes certain conditions under which computing the dimension of graph sums is easy and (2) constructs three infinitely-large classes of graphs that are minor minimal with respect to their dimension.