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A Method to Generate Multi-interval Pairwise Compatibility Graphs

Seemab Hayat, Naveed Ahmed Azam

TL;DR

To reduce the exponential tree search space, it is theoretically prove that for each multi-interval PCG there exists a tree whose internal vertices have degree exactly three, and an algorithm to enumerate such trees is developed.

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.

A Method to Generate Multi-interval Pairwise Compatibility Graphs

TL;DR

To reduce the exponential tree search space, it is theoretically prove that for each multi-interval PCG there exists a tree whose internal vertices have degree exactly three, and an algorithm to enumerate such trees is developed.

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.

Paper Structure

This paper contains 10 sections, 12 theorems, 11 figures, 6 tables, 7 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Some known results
  4. Proposed method to enumerate binary trees
  5. Proposed method to generate $k$-IPCGs
  6. Results and discussion
  7. Experimental results for graphs with eight vertices
  8. Experimental results for graphs with nine vertices
  9. Experimental results for with ten vertices
  10. Conclusion

Key Result

Lemma 1

azam2018enumerating There exist exactly seven non-PCGs with eight vertices.

Figures (11)

  • Figure 1: Phylogenetic tree of evolution of the DNA sequences tandy2009
  • Figure 2: (a) A PCG with five vertices; (b) An edge-weighted tree due to which the graph in (a) is PCG by considering the interval [5, 10] and the identity correspondence
  • Figure 3: A binary tree with nine leaves
  • Figure 4: (a) A non-PCG proved by Azam et al. azam2018enumerating which is a 2-IPCG identified by Calamoneri et al. calamoneri2022all with its witness tree in (b) and corresponding intervals $I_{1}=[1,3], I_{2}=[5,6]$
  • Figure 5: Transformations to convert a non-binary tree into a binary tree
  • ...and 6 more figures

Theorems & Definitions (15)

  • Lemma 1
  • Lemma 2
  • Lemma 3
  • Lemma 4
  • Lemma 5
  • Lemma 6
  • Lemma 7
  • proof
  • Lemma 8
  • proof
  • ...and 5 more