A Method to Generate Multi-interval Pairwise Compatibility Graphs
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866914974101667840 |
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| author | Hayat, Seemab Azam, Naveed Ahmed |
| author_facet | Hayat, Seemab Azam, Naveed Ahmed |
| contents | Reconstruction of evolutionary relationships between species is an important topic in the field of computational biology. Pairwise compatibility graphs (PCGs) are used to model such relationships. A graph is a PCG if its edges can be represented by the distance between the leaves of an edge-weighted tree within a fixed interval. If the number of intervals is more than one, then the graph with such a tree representation is called a multi-interval PCG. The aim of this paper is to generate all multi-interval PCGs with a given number of vertices. For this purpose, we propose a method to generate almost all multi-interval PCGs corresponding to a given tree by randomly assigning edge weights and selecting typical intervals. To reduce the exponential tree search space, we theoretically prove that for each multi-interval PCG there exists a tree whose internal vertices have degree exactly three, and developed an algorithm to enumerate such trees. The proposed method is applied to enumerate all two-interval PCGs with up to ten vertices. Our computational results establish that all graphs with up to ten vertices are 2-IPCGs, making significant progress towards the open problem of determining whether a non-2-IPCG exists with fewer than 135 vertices. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_10525 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A Method to Generate Multi-interval Pairwise Compatibility Graphs Hayat, Seemab Azam, Naveed Ahmed Discrete Mathematics Reconstruction of evolutionary relationships between species is an important topic in the field of computational biology. Pairwise compatibility graphs (PCGs) are used to model such relationships. A graph is a PCG if its edges can be represented by the distance between the leaves of an edge-weighted tree within a fixed interval. If the number of intervals is more than one, then the graph with such a tree representation is called a multi-interval PCG. The aim of this paper is to generate all multi-interval PCGs with a given number of vertices. For this purpose, we propose a method to generate almost all multi-interval PCGs corresponding to a given tree by randomly assigning edge weights and selecting typical intervals. To reduce the exponential tree search space, we theoretically prove that for each multi-interval PCG there exists a tree whose internal vertices have degree exactly three, and developed an algorithm to enumerate such trees. The proposed method is applied to enumerate all two-interval PCGs with up to ten vertices. Our computational results establish that all graphs with up to ten vertices are 2-IPCGs, making significant progress towards the open problem of determining whether a non-2-IPCG exists with fewer than 135 vertices. |
| title | A Method to Generate Multi-interval Pairwise Compatibility Graphs |
| topic | Discrete Mathematics |
| url | https://arxiv.org/abs/2410.10525 |