A Polynomial Kernel for Face Cover on Non-Embedded Planar Graphs

TL;DR

This paper introduces the first polynomial kernel for the Face Cover Number problem on planar graphs without a fixed embedding, using a bottom-up approach on SPR-trees to handle face boundary challenges.

Contribution

It presents a novel polynomial kernel for Face Cover Number on planar graphs without a predefined embedding, advancing kernelization techniques in graph theory.

Findings

First polynomial kernel for Face Cover Number on planar graphs without fixed embedding

Develops a bottom-up approach on SPR-trees to preserve face cover properties

Overcomes previous limitations of fixed embedding assumptions

Abstract

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.