Burnability of double spiders and path forests
•Discrete-time deterministic graph algorithm modelling the spread of social contagion.•Tight bounds on the burning numbers of double spiders and path forests.•Well-burnability of path forests with sufficiently long paths.•Progress towards a conjecture stronger than the burning number conjecture. The...
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| Published in: | Applied mathematics and computation Vol. 438; p. 127574 |
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| Main Authors: | , |
| Format: | Journal Article |
| Language: | English |
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01.02.2023
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| ISSN: | 0096-3003, 1873-5649 |
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| Abstract | •Discrete-time deterministic graph algorithm modelling the spread of social contagion.•Tight bounds on the burning numbers of double spiders and path forests.•Well-burnability of path forests with sufficiently long paths.•Progress towards a conjecture stronger than the burning number conjecture.
The burning number of a graph can be used to measure the spreading speed of contagion in a network. The burning number conjecture is arguably the main unresolved conjecture related to this graph parameter, which can be settled by showing that every tree of order m2 has burning number at most m. This is known to hold for many classes of trees, including spiders - trees with exactly one vertex of degree greater than two. In fact, it has been verified that certain spiders of order slightly larger than m2 also have burning numbers at most m, a result that has then been conjectured to be true for all trees. The first focus of this paper is to verify this slightly stronger conjecture for double spiders - trees with two vertices of degrees at least three and they are adjacent. Our other focus concerns the burning numbers of path forests, a class of graphs in which their burning numbers are naturally related to that of spiders and double spiders. Here, our main result shows that a path forest of order m2 with a sufficiently long shortest path has burning number exactly m, the smallest possible for any path forest of the same order. |
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| AbstractList | •Discrete-time deterministic graph algorithm modelling the spread of social contagion.•Tight bounds on the burning numbers of double spiders and path forests.•Well-burnability of path forests with sufficiently long paths.•Progress towards a conjecture stronger than the burning number conjecture.
The burning number of a graph can be used to measure the spreading speed of contagion in a network. The burning number conjecture is arguably the main unresolved conjecture related to this graph parameter, which can be settled by showing that every tree of order m2 has burning number at most m. This is known to hold for many classes of trees, including spiders - trees with exactly one vertex of degree greater than two. In fact, it has been verified that certain spiders of order slightly larger than m2 also have burning numbers at most m, a result that has then been conjectured to be true for all trees. The first focus of this paper is to verify this slightly stronger conjecture for double spiders - trees with two vertices of degrees at least three and they are adjacent. Our other focus concerns the burning numbers of path forests, a class of graphs in which their burning numbers are naturally related to that of spiders and double spiders. Here, our main result shows that a path forest of order m2 with a sufficiently long shortest path has burning number exactly m, the smallest possible for any path forest of the same order. |
| ArticleNumber | 127574 |
| Author | Teh, Wen Chean Tan, Ta Sheng |
| Author_xml | – sequence: 1 givenname: Ta Sheng orcidid: 0000-0002-5739-3242 surname: Tan fullname: Tan, Ta Sheng email: tstan@um.edu.my organization: Institute of Mathematical Sciences, Faculty of Science, Universiti Malaya, 50603 Kuala Lumpur, Malaysia – sequence: 2 givenname: Wen Chean orcidid: 0000-0001-8424-9820 surname: Teh fullname: Teh, Wen Chean email: dasmenteh@usm.my organization: School of Mathematical Sciences, Universiti Sains Malaysia, 11800 USM, Malaysia |
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| Keywords | 91D30 Burning number conjecture 05C85 Path forest Double spider Spread of social contagion 68R10 Graph algorithm |
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| SubjectTerms | Burning number conjecture Double spider Graph algorithm Path forest Spread of social contagion |
| Title | Burnability of double spiders and path forests |
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