Fragile minor-monotone parameters under random edge perturbation
Dong Yeap Kang, Mihyun Kang, Jaehoon Kim, Sang-il Oum
TL;DR
This work analyzes how a base graph $H$ on $n$ vertices with maximum degree $\Delta$ becomes structurally rich when augmented by a small random graph $G(n,p)$ to form $R=H\cup G(n,p)$. The authors prove that key minor-monotone parameters—tree-width $tw(R)$, genus $g(R)$, and Hadwiger number $h(R)$—can grow substantially beyond their base values with high probability, quantifying the growth in terms of $n$, $p$, and $\Delta$: $tw(R)=\Omega\left(tw(H)+\min\left(\frac{n^2 p}{\Delta},\;n\right)\right)$, $g(R)=\Omega\left(g(H)+\min\left((\frac{n^2 p}{\Delta})^2,\;n^2 p\right)\right)$, and $h(R)=\Omega\left(h(H)+\min\left(\sqrt{\frac{n^2 p}{\log\Delta}},\;\frac{n^2 p}{\Delta\sqrt{\log\Delta}}\right)\right)$. A central tool is a boosting lemma showing that adding a small amount of randomness can elevate minor-monotone parameters via an embedding of a moderately dense random graph $G(m,q)$ as a minor. The paper also develops sharpness results, extensions to base graphs with small path cover number, and a detailed proof framework for constructing the required vertex-disjoint connected subgraphs. Overall, the findings reveal a robust fragility phenomenon for these graph parameters under random edge perturbations, with implications for long cycles, forest minors, and related structural properties.
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 $H$.