Edgewise strongly shellable clutters

TL;DR

This paper investigates the strong shellability properties of clutters and graphs, establishing new results for chordal and bipartite graphs, and introduces the concept of edgewise strong shellability in these structures.

Contribution

It proves that the complement of chordal clutters is edgewise strongly shellable and characterizes edgewise strong shellability in bipartite graphs and graphs derived from trees.

Findings

Complement of chordal clutters is edgewise strongly shellable.

The generic graph of a tree is bi-strongly shellable.

Characterization of edgewise strongly shellable bipartite graphs.

Abstract

When $\mathcal{C}$ is a chordal clutter in the sense of Woodroofe or Emtander, we show that the complement clutter is edgewise strongly shellable. When $\mathcal{C}$ is indeed a finite simple graph, we study various characterizations of chordal graphs from the point of view of strong shellability. In particular, the generic graph $G_T$ of a tree is shown to be bi-strongly shellable. We also characterize edgewise strongly shellable bipartite graphs in terms of constructions from upward sequences. \end{abstract}