Given a planar graph, a subset of its vertices called terminals, and $k \in \mathbb{N}$, the Face Cover Number problem asks whether the terminals lie on the boundaries of at most $k$ faces of some embedding of the input graph. When a plane graph is given in the input, the problem is known to have a polynomial kernel~\cite{GarneroST17}. In this paper, we present the first polynomial kernel for Face Cover Number when the input is a planar graph (without a fixed embedding). Our approach overcomes the challenge of not having a predefined set of face boundaries by building a kernel bottom-up on an SPR-tree while preserving the essential properties of the face cover along the way.

翻译：给定一个平面图、其顶点的一个子集（称为终端）以及 $k \in \mathbb{N}$，面覆盖数问题询问：是否存在该输入图的某个嵌入，使得所有终端都位于至多 $k$ 个面的边界上。当输入中给定的是一个平面图（即已嵌入）时，已知该问题存在多项式核~\cite{GarneroST17}。在本文中，我们首次针对输入为平面图（无固定嵌入）的情况，为面覆盖数问题提出了一个多项式核。我们的方法通过自底向上地在 SPR 树上构建核，并在此过程中保持面覆盖的关键性质，从而克服了没有预定义面边界集合的挑战。