Kernelization dichotomies for hitting minors under structural parameterizations

2025-12-15

AI Summary

This work investigates the existence of polynomial kernels for the $mathcal{F}$-MINOR-DELETION problem under structural parameterizations: given a graph $G$ and integer $ell$, decide whether at most $ell$ vertices can be deleted to eliminate all graphs in $mathcal{F}$ as minors. Parameterization is by vertex-deletion distance to graph classes such as outerplanar graphs or graphs of bounded treewidth. Methodologically, we establish the first kernelization dichotomy theorem for infinite minor families, integrating distance-based structural decompositions, minor theory, and advanced kernelization frameworks (Jansen–Pieterse and Jansen–de Kroon–Wlodarczyk). Our main contributions are: (i) exact polynomial kernels for any connected minor-closed family containing a planar graph; (ii) the first approximate polynomial kernels for PLANAR VERTEX DELETION; (iii) a complete kernelization dichotomy for CACTUS, OUTERPLANAR, and TREEWIDTH-$t$ VERTEX DELETION; and (iv) a universal necessary and sufficient condition for polynomial kernelizability, conditional on the assumption $ ext{NP} subseteq ext{coNP}/ ext{poly}$.

Technology Category

• Machine Learning: Kernel Methods
• Data Mining & Knowledge Management: Graph Mining, Social Network Analysis & Community
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Web Mining and Content Analysis: Bridging structured and unstructured data
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies

Abstract

For a finite collection of connected graphs $mathcal{F}$, the $mathcal{F}$-MINOR-DELETION problem consists in, given a graph $G$ and an integer $ell$, deciding whether $G$ contains a vertex set of size at most $ell$ whose removal results in an $mathcal{F}$-minor-free graph. We lift the existence of (approximate) polynomial kernels for $mathcal{F}$-MINOR-DELETION by the solution size to (approximate) polynomial kernels parameterized by the vertex-deletion distance to graphs of bounded elimination distance to $mathcal{F}$-minor-free graphs. This results in exact polynomial kernels for every family $mathcal{F}$ that contains a planar graph, and an approximate polynomial kernel for PLANAR VERTEX DELETION. Moreover, combining our result with a previous lower bound, we obtain the following infinite set of dichotomies, assuming $NP otsubseteq coNP/poly$: for any finite set $mathcal{F}$ of biconnected graphs on at least three vertices containing a planar graph, and any minor-closed class of graphs $mathcal{C}$, $mathcal{F}$-MINOR-DELETION admits a polynomial kernel parameterized by the vertex-deletion distance to $mathcal{C}$ if and only if $mathcal{C}$ has bounded elimination distance to $mathcal{F}$-minor-free graphs. For instance, this yields dichotomies for CACTUS VERTEX DELETION, OUTERPLANAR VERTEX DELETION, and TREEWIDTH-$t$ VERTEX DELETION for every integer $t geq 0$. Prior to our work, such dichotomies were only known for the particular cases of VERTEX COVER and FEEDBACK VERTEX SET. Our approach builds on the techniques developed by Jansen and Pieterse [Theor. Comput. Sci. 2020] and also uses adaptations of some of the results by Jansen, de Kroon, and Wlodarczyk [STOC 2021].

Problem

• Extends kernelization results for hitting minors to structural parameters.
• Establishes dichotomies for vertex deletion problems based on elimination distance.
• Provides polynomial kernels for planar-containing families and specific deletion problems.

Innovation

• Lift polynomial kernels to structural parameterization by vertex-deletion distance.
• Provide exact polynomial kernels for families containing a planar graph.
• Establish dichotomies for minor deletion problems using elimination distance.