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arXiv · 2503.09230

Face covers and rooted minors in bounded genus graphs

Samuel Fiorini·Stefan Kober·Michał T. Seweryn·Abhinav Shantanam·Yelena Yuditsky
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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 Böhme and Mohar~\cite{BM02}. The higher genus case was claimed without a proof by Böhme, Kawarabayashi, Maharry and Mohar~\cite{BKMM08}.

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Samuel Fiorini, Stefan Kober, Michał T. Seweryn, Abhinav Shantanam, Yelena Yuditsky. 2025-03-12. Face covers and rooted minors in bounded genus graphs. https://arxiv.org/abs/2503.09230

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