Related papers: Slow graph bootstrap percolation II: Accelerating …
Given a fixed graph $H$ and an $n$-vertex graph $G$, the $H$-bootstrap percolation process on $G$ is defined to be the sequence of graphs $G_i$, $i\geq 0$ which starts with $G_0:=G$ and in which $G_{i+1}$ is obtained from $G_i$ by adding…
For graphs $H$, we study the extremal function $M_H(n)$ which is the maximum running time (until stabilisation) of an $H$-bootstrap percolation process on $n$ vertices. Building on previous work in the clique case $H=K_k$, we develop a…
The process of $H$-bootstrap percolation for a graph $H$ is a cellular automaton, where, given a subset of the edges of $K_n$ as initial set, an edge is added at time $t$ if it is the only missing edge in a copy of $H$ in the graph obtained…
For $k$-graphs $F$ and $H_0$ the $F$-bootstrap percolation process (or $F$-process) starting with $H_0$ is a sequence $(H_i)_{i\geq0}$ of $k$-graphs such that $H_{i+1}$ is obtained from $H_i$ by adding all those $e\in V(H_0)^{(k)}\setminus…
Consider the hypergraph bootstrap percolation process in which, given a fixed $r$-uniform hypergraph $H$ and starting with a given hypergraph $G_0$, at each step we add to $G_0$ all edges that create a new copy of $H$. We are interested in…
Graph bootstrap percolation is a simple cellular automaton introduced by Bollob\'as in 1968. Given a graph $H$ and a set $G \subseteq E(K_n)$ we initially "infect" all edges in $G$ and then, in consecutive steps, we infect every $e \in K_n$…
For $k$-graphs $F$ and $H_0$ the $F$-bootstrap percolation process (or $F$-process) starting with $H_0$ is a sequence $(H_i)_{i\geq0}$ of $k$-graphs such that $H_{i+1}$ is obtained from $H_i$ by adding all those $e\in V(H_0)^{(k)}\setminus…
Given $r\geq2$ and an $r$-uniform hypergraph $F$, the $F$-bootstrap process starts with an $r$-uniform hypergraph $H$ and, in each time step, every hyperedge which "completes" a copy of $F$ is added to $H$. The maximum running time of this…
Graph bootstrap percolation is a discrete-time process capturing the spread of a virus on the edges of $K_n$. Given an initial set $G\subseteq K_n$ of infected edges, the transmission of the virus is governed by a fixed graph $H$: in each…
Given two graphs $G$ and $H$, it is said that $G$ percolates in $H$-bootstrap process if one could join all the nonadjacent pairs of vertices of $G$ in some order such that a new copy of $H$ is created at each step. Balogh, Bollob\'as and…
Graph-bootstrap percolation, also known as weak saturation, was introduced by Bollob\'as in 1968. In this process, we start with initial "infected" set of edges $E_0$, and we infect new edges according to a predetermined rule. Given a graph…
Given a hypergraph $\mathcal{H}$, the $\mathcal{H}$-bootstrap process starts with an initial set of infected vertices of $\mathcal{H}$ and, at each step, a healthy vertex $v$ becomes infected if there exists a hyperedge of $\mathcal{H}$ in…
We study the following bootstrap percolation process: given a connected graph $G$, a constant $\rho \in [0, 1]$ and an initial set $A \subseteq V(G)$ of \emph{infected} vertices, at each step a vertex~$v$ becomes infected if at least a…
Bootstrap percolation on the random graph $G_{n,p}$ is a process of spread of "activation" on a given realization of the graph with a given number of initially active nodes. At each step those vertices which have not been active but have at…
In $\HH$-bootstrap percolation, a set $A \subset V(\HH)$ of initially 'infected' vertices spreads by infecting vertices which are the only uninfected vertex in an edge of the hypergraph $\HH$. A particular case of this is the $H$-bootstrap…
Bootstrap percolation on a graph is a deterministic process that iteratively enlarges a set of occupied sites by adjoining points with at least $\theta$ occupied neighbors. The initially occupied set is random, given by a uniform product…
Let $G_{n,p}^1$ be a superposition of the random graph $G_{n,p}$ and a one-dimensional lattice: the $n$ vertices are set to be on a ring with fixed edges between the consecutive vertices, and with random independent edges given with…
In the random $r$-neighbour bootstrap percolation process on a graph $G$, a set of initially infected vertices is chosen at random by retaining each vertex of $G$ independently with probability $p\in (0,1)$, and "healthy" vertices get…
Graph bootstrap percolation, introduced by Bollob\'as in 1968, is a cellular automaton defined as follows. Given a "small" graph $H$ and a "large" graph $G = G_0 \subseteq K_n$, in consecutive steps we obtain $G_{t+1}$ from $G_t$ by adding…
Bootstrap percolation on a graph with infection threshold $r\in \mathbb{N}$ is an infection process, which starts from a set of initially infected vertices and in each step every vertex with at least $r$ infected neighbours becomes…