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arXiv · 2410.10525

A Method to Generate Multi-interval Pairwise Compatibility Graphs

Seemab Hayat·Naveed Ahmed Azam
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Abstract

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.

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Seemab Hayat, Naveed Ahmed Azam. 2024-10-15. A Method to Generate Multi-interval Pairwise Compatibility Graphs. https://arxiv.org/abs/2410.10525

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