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Corpus ID: 235390376

Graphs that are minor minimal with respect to dimension

@inproceedings{Giardina2021GraphsTA,
title={Graphs that are minor minimal with respect to dimension},
author={Thomas John Giardina and Joel Foisy},
year={2021}
}

Erdős, Harary, and Tutte defined the dimension of a graph G as the smallest natural number n such that G can be embedded in R 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… Expand

In 1990, Y. Colin de Verdi ere introduced a new graph parameter (G), based on spectral properties of matrices associated with G. He showed that (G) is monotone under taking minors and that planarity… Expand

For any undirected graph G, let μ(G) be the graph parameter introduced by Colin de Verdiere. In this paper we show that μ(G) ≤ 4 if and only if G is linklessly embeddable (in R). This forms a… Expand

The main tools are a geometric formulation of the invariant, and constructing representations of graphs by spheres, related to the classical result of Koebe about representing planar graphs by touching disks, which show that such sphere representations characterize outerplanar and planar Graphs.Expand