A graph G is said to be chordal if it has no induced cycles of length four or more. In a recent preprint Culbertson, Guralnik, and Stiller give a new characterization of chordal graphs in terms of sequences of what they call ‘edge-erasures’. We show that these moves are in fact equivalent to a linear quotient ordering on ${\mathrm{I}}_{\bar{\mathrm{G}}}$, the edge ideal of the complement graph. Known results imply that ${\mathrm{I}}_{\bar{\mathrm{G}}}$ has linear quotients if and only if G is chordal, and hence this recovers an algebraic proof of their characterization. We investigate higher-dimensional analogues of this result, and show that in fact linear quotients for more general circuit ideals of d-clutters can be characterized in terms of removing exposed circuits in the complement clutter. Restricting to properly exposed circuits can be characterized by a homological condition. This leads to a notion of higher dimensional chordal clutters which borrows from commutative algebra and simple homotopy theory. The interpretation of linear quotients in terms of shellability of simplicial complexes also has applications to a conjecture of Simon regarding the extendable shellability of k-skeleta of simplices. Other connections to combinatorial commutative algebra, chordal complexes, and hierarchical clustering algorithms are explored.

如果曲线图G没有长度为四个或更多的诱导周期，则称该曲线图为弦。在最近的预印本中，Culbertson，Guralnik和Stiller根据所谓的“边缘擦除”的序列对弦图进行了新的表征。我们表明，这些移动实际上等效于上的线性商阶${\mathrm{一世}}_{\stackrel{〜}{\mathrm{G}}}$，补图的边缘理想值。已知结果暗示${\mathrm{一世}}_{\stackrel{〜}{\mathrm{G}}}$当且仅当G为弦时，才具有线性商，因此，这恢复了其表征的代数证明。我们研究了此结果的高维类似物，并表明，实际上，对于d杂波的更一般的电路理想而言，线性商可以通过去除补码杂波中的裸露电路来表征。限于适当暴露的电路可以通过同源条件来表征。这导致了更高维的弦杂波的概念，该概念借鉴了可交换代数和简单的同伦理论。关于单纯形配合物的可溶性的线性商的解释也适用于西蒙的一个猜想，即关于单纯形的k-skeleta的可扩展性。与组合交换代数，和弦复数的其他联系，