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Elimination distance to bounded degree on planar graphs

Alexander Lindermayr, Sebastian Siebertz, Alexandre Vigny

TL;DR

The article establishes fixed-parameter tractability for computing the elimination distance to bounded-degree graphs on planar (and more generally $K_5$-minor-free) graphs. It fuses grid-minor theory, irrelevant-vertex techniques, and MSO/Courcelle machinery to reduce instances to bounded-treewidth cases or to grid-minor certificates, enabling efficient decisions about membership in $\mathscr{C}_{k,d}$. The work expands the algorithmic applicability of elimination-distance concepts beyond minor-closed classes and provides a concrete $f(k,d)\cdot n^c$-time framework, with planarity improving to $f(k,d)\cdot n^3$. This advances understanding of how global properties like elimination distance interact with restricted graph classes and offers practical pathways for related parameterized problems.

Abstract

We study the graph parameter elimination distance to bounded degree, which was introduced by Bulian and Dawar in their study of the parameterized complexity of the graph isomorphism problem. We prove that the problem is fixed-parameter tractable on planar graphs, that is, there exists an algorithm that given a planar graph $G$ and integers $d$ and $k$ decides in time $f(k,d)\cdot n^c$ for a computable function~$f$ and constant $c$ whether the elimination distance of $G$ to the class of degree $d$ graphs is at most $k$.

Elimination distance to bounded degree on planar graphs

TL;DR

The article establishes fixed-parameter tractability for computing the elimination distance to bounded-degree graphs on planar (and more generally -minor-free) graphs. It fuses grid-minor theory, irrelevant-vertex techniques, and MSO/Courcelle machinery to reduce instances to bounded-treewidth cases or to grid-minor certificates, enabling efficient decisions about membership in . The work expands the algorithmic applicability of elimination-distance concepts beyond minor-closed classes and provides a concrete -time framework, with planarity improving to . This advances understanding of how global properties like elimination distance interact with restricted graph classes and offers practical pathways for related parameterized problems.

Abstract

We study the graph parameter elimination distance to bounded degree, which was introduced by Bulian and Dawar in their study of the parameterized complexity of the graph isomorphism problem. We prove that the problem is fixed-parameter tractable on planar graphs, that is, there exists an algorithm that given a planar graph and integers and decides in time for a computable function~ and constant whether the elimination distance of to the class of degree graphs is at most .

Paper Structure

This paper contains 4 sections, 10 theorems, 2 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Recent advances.
  3. Preliminaries
  4. The proof

Key Result

Theorem 1.1

There is an algorithm that for a $K_5$-minor-free input graph $G$ with $n$ vertices and integers $k,d$, decides in time $f(k,d)\cdot n^c$ whether $G$ belongs to $\mathscr{C}_{k,d}$, where $f$ is a computable function and $c$ is a constant. In case $G$ is $K_5$- and $K_{3,3}$-minor-free (that is, $G$

Figures (1)

  • Figure 1: Construction of a $K_5$ minor as soon as a branch set (here $H_1$) connected to a branch set (here $H_2$) that is at distance more than 2 in the grid book15parameterized-algorithms. The blue part marks the "outer part" of the border of thickness $2$, the "inner part" decomposes into a green, yellow and gray part.

Theorems & Definitions (12)

  • Theorem 1.1: Main result
  • Definition 2.1: Definition 4.2 of bulian2016graph
  • Proposition 2.2: Proposition 4.3 in bulian2016graph
  • Lemma 2.3
  • Theorem 2.1: Excluded Grid Theorem
  • Theorem 2.2: Planar Excluded Grid Theorem, see Theorem 7.23 of book15parameterized-algorithms
  • Theorem 2.3: Courcelle's Theorem courcelle1990monadic
  • Lemma 3.1
  • Corollary 3.2: of \ref{['lem-unbreak-weak']}
  • Definition 3.3
  • ...and 2 more