Spreading in graphs
Bo\v{s}tjan Bre\v{s}ar, Tanja Dravec, Aysel Erey, Jaka Hed\v{z}et

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 are contaminated initially, before the process starts. By the -forcing rule, a contaminated vertex having at most uncontaminated neighbors enforces all the neighbors to become contaminated, while by the -percolation rule, an uncontaminated vertex becomes contaminated if at least of its neighbors are contaminated. In this paper, we consider sets that are at the same time -forcing sets and -percolating sets, and call them -spreading sets. Given positive integers and , the minimum cardinality of a -spreading set in is a -spreading number, , of . While -forcing sets have been studied in a dozen of papers, the decision…
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Taxonomy
TopicsComplex Network Analysis Techniques · Graph theory and applications · Stochastic processes and statistical mechanics
