A $k$-stack layout (or $k$-page book embedding) of a graph consists of a total order of the vertices, and a partition of the edges into $k$ sets of non-crossing edges with respect to the vertex order. The stack number of a graph is the minimum $k$ such that it admits a $k$-stack layout. In this paper we study a long-standing problem regarding the stack number of planar directed acyclic graphs (DAGs), for which the vertex order has to respect the orientation of the edges. We investigate upper and lower bounds on the stack number of several families of planar graphs: We prove constant upper bounds on the stack number of single-source and monotone outerplanar DAGs and of outerpath DAGs, and improve the constant upper bound for upward planar 3-trees. Further, we provide computer-aided lower bounds for upward (outer-) planar DAGs.

中文翻译：

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关于具有恒定栈数的平面 DAG 族

图的 $k$-stack 布局（或 $k$-page book 嵌入）由顶点的总顺序和将边划分为与顶点相关的 $k$ 组非交叉边组成命令。图的堆栈数是最小的 $k$，这样它就可以使用 $k$-stack 布局。在本文中，我们研究了一个长期存在的关于平面有向无环图 (DAG) 堆栈数的问题，其中顶点顺序必须尊重边的方向。我们研究了几个平面图族的堆栈数的上限和下限：我们证明了单源和单调外平面 DAG 和外径 DAG 的堆栈数的恒定上限，并改进了向上平面 3-树木。此外，我们为向上（外）平面 DAG 提供了计算机辅助下界。