Lattice Minors and Eulerian Posets

TL;DR

This paper introduces the concept of minor posets derived from graph minors and polymatroids, proving their Eulerian property and establishing bounds on their cd-indices, thus advancing the understanding of their combinatorial structure.

Contribution

It defines minor posets for graphs and polymatroids, proves they are Eulerian face posets of regular CW spheres, and establishes bounds on their cd-indices.

Findings

Minor posets are isomorphic to face posets of regular CW spheres.

Minor posets are Eulerian.

Established tight bounds for cd-indices of minor posets.

Abstract

We introduce posets of simple vertex labeled minors of graphs and a generalization to the level of polymatroids, collectively termed minor posets. We show that any minor poset is isomorphic to the face poset of a regular CW sphere, and in particular, is Eulerian. We establish cd-index inequalities induced by strong maps, a tight upper bound for cd-indices of minor posets and a tight lower bound for cd-indices of minor posets arising from lattices of maximal length.