| Title: | On polluted bootstrap percolation in Cartesian grids |
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| Authors: | ID Brešar, Boštjan (Author)
ID Hedžet, Jaka (Author)
ID Henning, Michael A. (Author) |
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| Files: |  PDF - Presentation file, download (154,01 KB)
MD5: 739FCC048401FC40AA6FC30B1DB5B824
 URL - Source URL, visit https://ajc.maths.uq.edu.au/pdf/96/ajc_v96_p027.pdf
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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 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 continue the study of polluted bootstrap percolation introduced and studied by Gravner and McDonald [J. Stat Physics 87 (1997) 915-927] where in this variant some vertices are permanently in the non-infected state. We study an extremal (combinatorial) version of the bootstrap percolation problem in a polluted environment, where our main focus is on the class of grid graphs, that is, the Cartesian product $P_m \Box P_n$ of two paths $P_m$ and $P_n$ on $m$ and $n$ vertices, respectively. Given a number of polluted vertices in a Cartesian grid we establish a closed formula for the minimum $2$-neighbor bootstrap percolation number of the polluted grid, and obtain a lower bound for the other extreme. |
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| Keywords: | bootstrap percolation, grid, polluted environment, vertex deleted subgraph |
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| Publication status: | Published |
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| Publication version: | Version of Record |
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| Publication date: | 01.10.2026 |
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| Year of publishing: | 2026 |
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| Number of pages: | str. 27-37 |
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| Numbering: | Vol. 96, part 1 |
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| PID: | 20.500.12556/DiRROS-31905  |
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| UDC: | 519.17 |
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| ISSN on article: | 2202-3518 |
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| COBISS.SI-ID: | 287971075 |
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| Publication date in DiRROS: | 17.08.2026 |
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| Views: | 40 |
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| Downloads: | 19 |
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