Algorithmic Applications of Tyshkevich's Graph Decomposition: A Primer and a Toolkit

TL;DR

This paper reviews Tyshkevich's graph decomposition theorem for unigraphs, explaining how it enables linear-time computation of graph parameters and providing a toolkit for practical implementation.

Contribution

It offers an accessible overview of Tyshkevich's decomposition theorem and introduces a toolkit implementing the related algorithms for unigraph analysis.

Findings

Linear-time algorithms for graph parameter computation

Complete classification of basic unigraphs

Practical toolkit for graph decomposition

Abstract

A graph that is completely determined by its degree sequence is called a unigraph. In 2000, Regina Tyshkevich published one of the most important papers on unigraphs. There are two parts to the paper: a decomposition theorem that describes how every graph can be broken into a sequence of basic graphs and a complete classification of all basic unigraphs. Together, they reveal how every unigraph is constructed. We provide an informal overview of Tyshkevich's results and show how they enable the computation of various graph parameters of unigraphs in linear time. We also created a toolkit (https://chelseal11.github.io/tyshkevich_decomposition_toolkit/) that implements the algorithms described in this write-up.