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

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.