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

2026-01-07

arXiv.org

AI Summary

This study addresses the Face Cover Number problem on plane graphs without a fixed embedding: determining whether a given set of terminal vertices can be covered by the boundaries of at most $k$ faces. The paper presents the first polynomial kernelization algorithm for this problem in the non-embedded setting. By constructing an SPR-tree bottom-up and leveraging the combinatorial structural properties of planar graphs, the method effectively handles the uncertainty inherent in face boundaries while preserving the essential characteristics relevant to face covering. This work overcomes the previous limitation that kernelization results were only available for graphs with a fixed embedding, establishing the first polynomial kernel for unembedded planar graphs and laying a theoretical foundation for future efficient parameterized algorithms.

Technology Category

• Machine Learning: Kernel Methods
• Search and Optimization: Combinatorial Optimization
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories

Application Category

• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics

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.

Problem

• Face Cover
• Planar Graph
• Polynomial Kernel
• Graph Embedding
• Parameterized Complexity

Innovation

• polynomial kernel
• face cover
• planar graph
• SPR-tree
• graph embedding