This paper builds upon the framework of \emph{Morse sequences}, a simple and effective approach to discrete Morse theory. A Morse sequence on a simplicial complex consists of a sequence of nested subcomplexes generated by expansions and fillings-two operations originally introduced by Whitehead. Expansions preserve homotopy, while fillings introduce critical simplexes that capture essential topological features. We extend the notion of Morse sequences to \emph{stacks}, which are monotonic functions defined on simplicial complexes, and define \emph{Morse sequences on stacks} as those whose expansions preserve the homotopy of all sublevel sets. This extension leads to a generalization of the fundamental collapse theorem to weighted simplicial complexes. Within this framework, we focus on a refined class of sequences called \emph{flooding sequences}, which exhibit an ordering behavior similar to that of classical watershed algorithms. Although not every Morse sequence on a stack is a flooding sequence, we show that the gradient vector field associated with any Morse sequence can be recovered through a flooding sequence. Finally, we present algorithmic schemes for computing flooding sequences using cosimplicial complexes.

翻译：本文建立在 \emph{Morse 序列} 的框架之上，这是离散 Morse 理论中一种简单而有效的方法。单纯复形上的 Morse 序列由一系列嵌套的子复形构成，这些子复形通过扩张与填充生成——这两种运算最初由 Whitehead 引入。扩张保持同伦性，而填充则引入捕获基本拓扑特征的关键单形。我们将 Morse 序列的概念推广到 \emph{堆栈}（即在单纯复形上定义的单调函数），并将 \emph{堆栈上的 Morse 序列} 定义为那些扩张能保持所有子水平集同伦性的序列。这一推广将基本坍缩定理推广到了加权单纯复形。在此框架内，我们专注于一类称为 \emph{淹没序列} 的精细序列，它们展现出与经典分水岭算法类似的排序行为。尽管并非所有堆栈上的 Morse 序列都是淹没序列，但我们证明了与任意 Morse 序列相关的梯度向量场均可通过一个淹没序列恢复。最后，我们提出了利用余单纯复形计算淹没序列的算法方案。