# Upward-closed hereditary families in the dominance order

## TL;DR

This paper explores the properties of upward-closed hereditary families within dominance orders of degree sequences, characterizing when certain graph classes are dominance monotone based on forbidden induced subgraphs.

## Contribution

It introduces the concept of dominance monotone graph classes and provides necessary conditions and classifications for sets of graphs of size up to three.

## Key findings

- Threshold and split graph degree sequences form upward-closed sets.
- Necessary conditions for dominance monotonicity are established.
- Classification of dominance monotone sets of order at most 3.

## Abstract

The majorization relation orders the degree sequences of simple graphs into posets called dominance orders. As shown by Ruch and Gutman (1979) and Merris (2002), the degree sequences of threshold and split graphs form upward-closed sets within the dominance orders they belong to, i.e., any degree sequence majorizing a split or threshold sequence must itself be split or threshold, respectively. Motivated by the fact that threshold graphs and split graphs have characterizations in terms of forbidden induced subgraphs, we define a class $\mathcal{F}$ of graphs to be dominance monotone if whenever no realization of $e$ contains an element $\mathcal{F}$ as an induced subgraph, and $d$ majorizes $e$, then no realization of $d$ induces an element of $\mathcal{F}$. We present conditions necessary for a set of graphs to be dominance monotone, and we identify the dominance monotone sets of order at most 3.