| 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: |  On_polluted_bootstrap_percolation-Bresar-2026.pdf (155,00 KB)
MD5: 117DAD6A5E6BA694D9176AFB7D06C273
 https://ajc.maths.uq.edu.au/pdf/96/ajc_v96_p027.pdf
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| Language: | English |
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| Work type: | Scientific work |
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| Typology: | 1.01 - Original Scientific Article |
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| Organization: | FNM - Faculty of Natural Sciences and Mathematics
FKKT - Faculty of Chemistry and Chemical Engineering
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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: | ojačano pronicanje, mreža, onesnaženo okolje, podgraf z odstranjenim vozliščem, 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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| Submitted for review: | 23.06.2025 |
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| Year of publishing: | 2026 |
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| Number of pages: | str. 27-37 |
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| Numbering: | Letn. 96, št. 1 |
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| PID: | 20.500.12556/DKUM-99516  |
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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 DKUM: | 20.08.2026 |
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| Views: | 248 |
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| Downloads: | 0 |
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| Metadata: |    |
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| Categories: | Misc.
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