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arXiv · 2609.01851

Minimum Weakly Saturated Graphs and Bootstrap Percolation in General Host Graphs

Roman Vasquez
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Abstract

A graph $G$ is weakly $H$-saturated if one can obtain $K_n$ by adding one edge to $G$ at a time, where each additional edge creates at least one new copy of $H$. The minimum number of edges needed for a weakly $H$-saturated graph $G$ of order $n$ is known as the weak saturation number of $H$, written $wsat(n,H)$. A graph $G$ is minimum weakly saturated if $wsat(n,G)=|E(G)|-1$ for some value of $n$. We explore classes of minimum weakly saturated graphs and their connection to the $H$-bootstrap percolation process, as well as weak saturation in a more general setting than the complete graph.

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Roman Vasquez. 2026-09-01. Minimum Weakly Saturated Graphs and Bootstrap Percolation in General Host Graphs. https://arxiv.org/abs/2609.01851

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