Upward-closed hereditary families in the dominance order

Abstract

The majorization relation orders the degree sequences of simple graphs intoposets called dominance orders. As shown by Ruch and Gutman (1979) and Merris(2002), the degree sequences of threshold and split graphs form upward-closedsets within the dominance orders they belong to, i.e., any degree sequencemajorizing a split or threshold sequence must itself be split or threshold,respectively. Motivated by the fact that threshold graphs and split graphs havecharacterizations 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 presentconditions necessary for a set of graphs to be dominance monotone, and weidentify the dominance monotone sets of order at most 3.Comment: 19 pages, 5 figures. Final accepted versio