Bootstrap percolation in strong products of graphs

TL;DR

This paper investigates the minimum initial infected vertices needed for complete infection spread in strong product graphs under bootstrap percolation, revealing exact thresholds and large percolation numbers in various graph families.

Contribution

It provides exact formulas and dichotomies for percolation numbers in strong products of graphs, extending understanding of bootstrap percolation thresholds in complex graph structures.

Findings

For strong products of k graphs, m(G,r)=r when r≤2^{k-1} and |V(G)|≥r.

Existence of strong product families with arbitrarily large (r+1)-percolation numbers.

Bounds on m(G,r) for products where factors have at least three vertices.

Abstract

Given a graph $G$ and assuming that some vertices of $G$ are infected, the $r$-neighbor bootstrap percolation rule makes an uninfected vertex $v$ infected if $v$ has at least $r$ infected neighbors. The $r$-percolation number, $m(G,r)$, of $G$ is the minimum cardinality of a set of initially infected vertices in $G$ such that after continuously performing the $r$-neighbor bootstrap percolation rule each vertex of $G$ eventually becomes infected. In this paper, we consider percolation numbers of strong products of graphs. If $G$ is the strong product $G_1\boxtimes \cdots \boxtimes G_k$ of $k$ connected graphs, we prove that $m(G,r)=r$ as soon as $r\le 2^{k-1}$ and $|V(G)|\ge r$. As a dichotomy, we present a family of strong products of $k$ connected graphs with the $(2^{k-1}+1)$-percolation number arbitrarily large. We refine these results for strong products of graphs in which at least two factors have at least three vertices. In addition, when all factors $G_i$ have at least three vertices we prove that $m(G_1 \boxtimes \dots \boxtimes G_k,r)\leq 3^{k-1} -k$ for all $r\leq 2^k-1$, and we again get a dichotomy, since there exist families of strong products of $k$ graphs such that their $2^{k}$-percolation numbers are arbitrarily large. While $m(G\boxtimes H,3)=3$ if both $G$ and $H$ have at least three vertices, we also characterize the strong prisms $G\boxtimes K_2$ for which this equality holds. Some of the results naturally extend to infinite graphs, and we briefly consider percolation numbers of strong products of two-way infinite paths.