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Face covers and rooted minors in bounded genus graphs

Samuel Fiorini, Stefan Kober, Michał T. Seweryn, Abhinav Shantanam, Yelena Yuditsky

TL;DR

The paper studies when rooted graphs embedded on a fixed surface admit small face covers, identifying rooted $K_{2,t}$-minors as the main obstructions. It develops a planar approach using Schnyder embeddings and poset arguments to prove an $O(t^4)$ bound for the planar case, and then extends to bounded-genus surfaces by planarization and surface-covers, achieving $O(g t^4)$ under large face-width. It also provides lower-bound constructions (windmill and bagel) showing the bounds can be nontrivial, and handles the projective plane separately before addressing the general genus case via Yu’s planarization framework. The results provide a structural link between topological embedding properties and the existence of small face covers, with potential algorithmic implications for related optimization problems on embedded graphs.

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}.

Face covers and rooted minors in bounded genus graphs

TL;DR

The paper studies when rooted graphs embedded on a fixed surface admit small face covers, identifying rooted -minors as the main obstructions. It develops a planar approach using Schnyder embeddings and poset arguments to prove an bound for the planar case, and then extends to bounded-genus surfaces by planarization and surface-covers, achieving under large face-width. It also provides lower-bound constructions (windmill and bagel) showing the bounds can be nontrivial, and handles the projective plane separately before addressing the general genus case via Yu’s planarization framework. The results provide a structural link between topological embedding properties and the existence of small face covers, with potential algorithmic implications for related optimization problems on embedded graphs.

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 -connected, rooted graph that has no rooted minor and is embedded in a surface of Euler genus , has a face cover whose size is upper-bounded by some function of and , provided that the face-width of the embedding is large enough in terms of . In the planar case, we prove an unconditional 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}.

Paper Structure

This paper contains 12 sections, 17 theorems, 1 equation, 8 figures.

Table of Contents

  1. Introduction
  2. Background
  3. Graph terminology
  4. Schnyder embeddings
  5. Graphs on surfaces
  6. The planar case
  7. The bounded genus case
  8. The projective planar case
  9. The general case
  10. Construction of an auxiliary tree $T$.
  11. Using $T$ to find a surface cover.
  12. Properties of the surface cover.

Key Result

Theorem 1

There exists a function $f_{thm:planar} : \mathbb{Z}_{\geqslant 1} \to \mathbb{Z}_{\geqslant 0}$ with $f_{thm:planar}(t) = O(t^4)$ such that the following holds. If $(G,R)$ is a $3$-connected plane rooted graph without a rooted $K_{2,t}$ minor, then $(G,R)$ admits a face cover of size at most $f_{th

Figures (8)

  • Figure 1: The windmill graph of parameter $t = 10$. Generalizing this construction gives a family of $3$-connected, plane rooted graphs with no rooted $K_{2,t}$ minor in which every face cover has size $\Omega(t^2)$. See \ref{['prop:windmill']} below for a proof.
  • Figure 2: The figure shows a bagel graph embedded in the torus, with $n = 14$ vertices, all included in the root set. The construction generalizes to every even $n \geqslant 6$ and gives a $4$-connected, $n$-vertex rooted graph $B_n$ embedded on the torus and without rooted $K_{2,t}$ minor for $t \geqslant 5$. However every face cover has size $\Omega(n)$. Notice that the face-width of $B_n$ is $2$. See \ref{['prop:bagel']} for details.
  • Figure 3: Cutting along cycle $C$ (two-sided case). To the left, graph $G$ and surface $\Sigma$ before cutting. To the right, graph $G'$ and surface with boundary $\Sigma'$ after cutting.
  • Figure 4: Roots $u$ and $v$ such that $u <_i v$, together with the three parallelograms with corner $u$ and the three parallelograms with corner $v$.
  • Figure 5: Set of roots $S$ such that $u_i = c$ where $c$ is a constant, shown with $\Pi_i(S)$ (red parallelograms) and $\Pi_{i-1}(V)$ where $V := \{v(u) : u \in S\}$ and $v(u)$ is the first vertex of $P(u)$ that has $v_i < c$ (green parallelograms).
  • ...and 3 more figures

Theorems & Definitions (33)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Proposition 4
  • Proposition 5: Mohar Moh97
  • Theorem 6: Bienstock and Dean BD92
  • proof : Proof of \ref{['thm:planar']}
  • Claim 1
  • Claim 2
  • Proposition 7
  • ...and 23 more