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.

翻译：完全由其度序列决定的图称为单图。2000年，Regina Tyshkevich发表了关于单图的最重要论文之一。该论文包含两部分：描述每个图如何分解为一系列基本图的分解定理，以及对所有基本单图的完整分类。二者共同揭示了每个单图的构造方式。本文对Tyshkevich的研究成果进行了非正式概述，并展示了如何利用这些成果在O(n)时间内计算单图的各种图参数。我们还开发了一个工具包（https://chelseal11.github.io/tyshkevich_decomposition_toolkit/），实现了本文所述的算法。