Largest and Smallest Minimal Percolating Sets in Trees

Abstract

Originally introduced by Chalupa, Leath and Reich for use in modeling disordered magnetic systems, $r$-bootstrap percolation is the following deterministic process on a graph. Given an initial infected set, vertices with at least $r$ infected neighbors are infected until no new vertices can be infected. A set percolates if it infects all the vertices of the graph, and a percolating set is minimal if no proper subset percolates. We consider minimal percolating sets in finite trees. We show that if $A$ is a minimal percolating set on a tree $T$ with $n$ vertices and $\ell$ vertices of degree less than $r$ (leaves in the case $r=2$), then $\frac{(r-1)n+1}{r} \leq |A| \leq \frac{rn+\ell}{r+1}$. Moreover, we show that the difference between the sizes of a largest and smallest minimal percolating sets is at most $\frac{(r-1)(n-1)}{r^2}$. Finally, we describe $O(n)$ algorithms for computing the largest (for $r=2$) and smallest (for $r \geq 2$) minimal percolating sets.

Largest Minimal Percolating Sets in Hypercubes under $2$-Bootstrap Percolation

“…For r = 2, a simple inductive argument shows that any minimal percolating set has size 2, since any minimal percolating set must eventually infect two adjacent vertices in a sub-AQ n−1 , which will percolate. However, using the "wasted" edge-counting technique from [13], we see that E(AQ 6 , 7) 14, so percolation is nontrivial for larger r. As this example shows, percolation depends heavily on the structure of the graph, as the Augmented Hypercube has only twice the edges of the standard hypercube, yet percolation is quite different.…”

“…asymptotically, making progress on a question posed by Bollobás. In [13], an algorithm is presented for finding m(T, r) and E(T, r) for all finite trees T , and it is shown that if T is a finite tree with ℓ leaves, m(T, r)…”

Deterministic bootstrap percolation on trees

“…Alternatively, instead of choosing our initial set randomly, we choose A 0 k (G) in order to insure that every vertex in a graph is eventually infected. While this deterministic approach seems to be less common historically, it has been considered in [9,10,12,13,22,23,24,25] and the appendix of [4]. We should also mention that in addition to the standard bootstrap percolation considered in this paper, there are also several variants.…”

Ore and Chvátal‐type degree conditions for bootstrap percolation from small sets

“…In addition, there has recently been interest in extremal problems concerning bootstrap percolation in various families of graphs .…”