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 improve the 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.

翻译：一个图的 $k$-栈布局（或 $k$-页书嵌入）由顶点的全序和边划分为 $k$ 个关于该顶点序的非交叉边集组成。图的栈数是使得其允许 $k$-栈布局的最小 $k$ 值。本文研究了一个关于平面有向无环图（DAG）栈数的长期未解问题，其中顶点序必须尊重边的方向。我们探究了若干平面图族栈数的上下界：改进了单源和单调外平面DAG以及外路径DAG栈数的常数上界，并改进了向上平面3-树栈数的常数上界。此外，我们还为向上（外）平面DAG提供了计算机辅助的下界。