Chordality of Clutters with Vertex Decomposable Dual and Ascent of Clutters

In this paper, we consider the generalization of chordal graphs to clutters proposed by Bigdeli, et al in J. Combin. Theory, Series A (2017). Assume that $\mathcal{C}$ is a $d$-dimensional uniform clutter. It is known that if $\mathcal{C}$ is chordal, then $I(\bar{\mathcal{C}})$ has a linear resolution over all fields. The converse has recently been rejected, but the following question which poses a weaker version of the converse is still open: "if $I(\bar{\mathcal{C}})$ has linear quotients, is $\mathcal{C}$ necessarily chordal?". Here, by introducing the concept of the ascent of a clutter, we split this question into two simpler questions and present some clues in support of an affirmative answer. In particular, we show that if $I(\bar{\mathcal{C}})$ is the Stanley-Reisner ideal of a simplicial complex with a vertex decomposable Alexander dual, then $\mathcal{C}$ is chordal.