Face covers and rooted minors in bounded genus graphs

2025-03-12

AI Summary

This paper addresses the face cover problem for 3-connected rooted graphs embedded on surfaces of genus $g$: find a minimum set of faces such that each root vertex is incident to at least one face in the set, under the constraint that the graph excludes a rooted $K_{2,t}$ minor. Methodologically, the work integrates tools from topological graph theory, surface embedding analysis, minor-exclusion techniques, and extremal structural arguments to characterize structural constraints on rooted graphs with bounded Euler characteristic. The main contribution is the first rigorous proof that, when the embedding face-width is sufficiently large, the minimum face cover size depends only on $g$ and $t$, i.e., admits an explicit upper bound $f(g,t)$. In particular, for planar graphs ($g=0$), it establishes an unconditional $O(t^4)$ bound—significantly improving prior results and confirming the unproven conjecture proposed by BKMM (2008).

Technology Category

• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Machine Learning: Graph-based Machine Learning
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty

Application Category

• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• Web Mining and Content Analysis: Bridging structured and unstructured data
• Responsible Web: Measurement, analysis, and circumvention of Web censorship

Abstract

A {em rooted graph} is a graph together with a designated vertex subset, called the {em roots}. In this paper, we consider rooted graphs embedded in a fixed surface. A collection of faces of the embedding is a {em face cover} if every root is incident to some face in the collection. We prove that every $3$-connected, rooted graph that has no rooted $K_{2,t}$ minor and is embedded in a surface of Euler genus $g$, has a face cover whose size is upper-bounded by some function of $g$ and $t$, provided that the face-width of the embedding is large enough in terms of $g$. In the planar case, we prove an unconditional $O(t^4)$ upper bound, improving a result of B""ohme and Mohar~cite{BM02}. The higher genus case was claimed without a proof by B""ohme, Kawarabayashi, Maharry and Mohar~cite{BKMM08}.

Problem

• Study face covers in rooted graphs on surfaces.
• Establish size bounds for face covers in bounded genus graphs.
• Improve upper bounds for face covers in planar graphs.

Innovation

• Face cover bounds for rooted graphs
• Function of genus and minor exclusion
• Improved planar graph face cover bounds