| Title: | Bootstrap percolation in strong products of graphs |
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| Authors: | ID Brešar, Boštjan (Author)
ID Hedžet, Jaka (Author) |
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| Files: |  PDF - Presentation file, download (649,39 KB)
MD5: 0221CDEFE7DA9103CF5B6F6C2C59DC4F
 URL - Source URL, visit https://www.combinatorics.org/ojs/index.php/eljc/article/view/v31i4p35
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| Language: | English |
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| Typology: | 1.01 - Original Scientific Article |
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| Organization: |  IMFM - Institute of Mathematics, Physics, and Mechanics
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| 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. |
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| Keywords: | bootstrap percolation, strong product of graphs, infinite path |
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| Publication status: | Published |
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| Publication version: | Version of Record |
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| Publication date: | 01.01.2024 |
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| Year of publishing: | 2024 |
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| Number of pages: | 22 str. |
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| Numbering: | Vol. 31, iss. 4, article no. P4.35 |
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| PID: | 20.500.12556/DiRROS-20844  |
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| UDC: | 519.17:519.2 |
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| ISSN on article: | 1077-8926 |
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| DOI: | 10.37236/11826 |
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| COBISS.SI-ID: | 215693827 |
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| Note: |
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| Publication date in DiRROS: | 20.11.2024 |
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| Views: | 89 |
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| Downloads: | 46 |
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