Upward-closed hereditary families in the dominance order

The majorization relation orders the degree sequences of simple graphs into posets called dominance orders. As shown by Hammer et al. and Merris, 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.Comment: 15 pages, 6 figure

Schubert problems with respect to osculating flags of stable rational curves

Given a point z in P^1, let F(z) be the osculating flag to the rational normal curve at point z. The study of Schubert problems with respect to such flags F(z_1), F(z_2), ..., F(z_r) has been studied both classically and recently, especially when the points z_i are real. Since the rational normal curve has an action of PGL_2, it is natural to consider the points (z_1, ..., z_r) as living in the moduli space of r distinct point in P^1 -- the famous M_{0,r}. One can then ask to extend the results on Schubert intersections to the compactification \bar{M}_{0,r}. The first part of this paper achieves this goal. We construct a flat, Cohen-Macaulay family over \bar{M}_{0,r}, whose fibers over M_{0,r} are isomorphic to G(d,n) and, given partitions lambda_1, ..., lambda_r, we construct a flat Cohen-Macualay family over \bar{M}_{0,r} whose fiber over (z_1, ..., z_r) in M_{0,r} is the intersection of the Schubert varieties indexed by lambda_i with respect to the osculating flags F(z_i). In the second part of the paper, we investigate the topology of the real points of our family, in the case that sum |lambda_i| = dim G(d,n). We show that our family is a finite covering space of \bar{M}_{0,r}, and give an explicit CW decomposition of this cover whose faces are indexed by objects from the theory of Young tableaux