Fragile minor-monotone parameters under random edge perturbation

TL;DR

This paper analyzes how adding a small number of random edges to a connected graph can cause significant increases in parameters like tree-width, genus, and Hadwiger number, regardless of the original graph's structure.

Contribution

It provides a quantitative understanding of the fragility of minor-monotone parameters under random edge perturbations in graphs.

Findings

Adding few random edges greatly increases tree-width, genus, and Hadwiger number.

Minor-monotone parameters are highly sensitive to random edge additions.

Results hold regardless of the initial graph's structure.

Abstract

We conduct a quantitative analysis of how many random edges need to be added to a base graph $H$ in order to significantly increase natural minor-monotone graph parameters of the resulting graph $R$. Specifically, we show that if $R$ is obtained from a connected graph $H$ by adding only a few random edges, the tree-width, genus, and Hadwiger number of $R$ become very large, irrespective of the structure of $H$.