A Method to Generate Multi-interval Pairwise Compatibility Graphs

TL;DR

This paper introduces a method to generate and enumerate multi-interval pairwise compatibility graphs (PCGs), specifically focusing on two-interval PCGs with up to ten vertices, advancing understanding of their structure and existence.

Contribution

It presents a novel algorithm to generate multi-interval PCGs by reducing the search space and proves that relevant trees can be simplified to degree-three internal vertices.

Findings

All graphs with up to ten vertices are 2-IPCGs.

The method significantly reduces the search space for generating such graphs.

Progress towards the open problem of non-2-IPCG existence with fewer than 135 vertices.

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.