The Hereditary Closure of the Unigraphs

TL;DR

This paper characterizes the hereditary closure of unigraphs, a class of graphs where all induced subgraphs are also unigraphs, using decomposition and degree sequence tools.

Contribution

It provides multiple characterizations of the hereditary closure of unigraphs, expanding understanding of their structural properties.

Findings

Characterization of the hereditary closure of unigraphs

Use of Tyshkevich decomposition and Rao's partial order

New criteria for unigraphs with all induced subgraphs also unigraphs

Abstract

A graph with degree sequence $\pi$ is a \emph{unigraph} if it is isomorphic to every graph that has degree sequence $\pi$. The class of unigraphs is not hereditary and in this paper we study the related hereditary class HCU, the hereditary closure of unigraphs, consisting of all graphs induced in a unigraph. We characterize the class HCU in multiple ways making use of the tools of a decomposition due to Tyshkevich and a partial order on degree sequences due to Rao. We also provide a new characterization of the class that consists of unigraphs for which all induced subgraphs are also unigraphs.