Spreading in graphs

TL;DR

This paper studies the properties of spreading processes in graphs, introduces the concept of $(p,q)$-spreading sets, and explores their computational complexity and bounds in different graph classes.

Contribution

It defines the $(p,q)$-spreading number, proves NP-completeness of related decision problems, and provides efficient algorithms and bounds for trees and grid graphs.

Findings

NP-completeness of the $(p,q)$-spreading number decision problem

Linear-time algorithm for trees

Explicit formulas for grid graphs

Abstract

Several concepts that model processes of spreading (of information, disease, objects, etc.) in graphs or networks have been studied. In many contexts, we assume that some vertices of a graph $G$ are contaminated initially, before the process starts. By the $q$-forcing rule, a contaminated vertex having at most $q$ uncontaminated neighbors enforces all the neighbors to become contaminated, while by the $p$-percolation rule, an uncontaminated vertex becomes contaminated if at least $p$ of its neighbors are contaminated. In this paper, we consider sets $S$ that are at the same time $q$-forcing sets and $p$-percolating sets, and call them $(p,q)$-spreading sets. Given positive integers $p$ and $q$, the minimum cardinality of a $(p,q)$-spreading set in $G$ is a $(p,q)$-spreading number, $\sigma_{(p,q)}(G)$, of $G$. While $q$-forcing sets have been studied in a dozen of papers, the decision version of the corresponding graph invariant has not been considered earlier, and we fill the gap by proving its NP-completeness. This, in turn, enables us to prove the NP-completeness of the decision version of the $(p,q)$-spreading number in graphs for an arbitrary choice of $p$ and $q$. Again, for every $p\in \mathbb{N}$ and $q\in\mathbb{N}\cup\{\infty\}$, we find a linear-time algorithm for determining the $(p,q)$-spreading number of a tree. In addition, we present a lower and an upper bound on the $(p,q)$-spreading number of a tree and characterize extremal families of trees. In the case of square grids, we combine some known results and new results on $(2,1)$-spreading and $(4,q)$-spreading to obtain $\sigma_{(p,q)}(P_m\Box P_n)$ for all $(p,q)\in (\mathbb{N}\setminus\{3\})\times (\mathbb{N}\cup\{\infty\})$ and all $m,n\in\mathbb{N}$.