Graph modification of bounded size to minor-closed classes as fast as vertex deletion

2025-04-23

AI Summary

This paper studies a generalized graph modification problem—“$mathcal{L}$-replacement into $mathcal{H}$”: given a graph $G$, a minor-closed graph class $mathcal{H}$, and a local replacement rule $mathcal{L}$, decide whether $G$ can be transformed into a graph in $mathcal{H}$ via at most $k$ operations, each replacing an induced subgraph $H_1$ with some $H_2 in mathcal{L}(H_1)$. This framework unifies classical problems including vertex deletion, edge editing/contraction, and subgraph completion. We present the first fixed-parameter tractable (FPT) algorithm for this general model with reasonable parameter dependence: for any minor-closed $mathcal{H}$, the running time is $2^{mathrm{poly}(k)} cdot |V(G)|^2$; if $G$ embeds on a surface of genus at most $g$, it improves to $2^{O(k^9)} cdot |V(G)|^2$. Our key technical innovation extends structural results for vertex deletion on minor-closed classes to non-local, structure-aware replacement operations—enabling decomposition and replacement strategies guided by graph minors theory.

Technology Category

• Machine Learning: Graph-based Machine Learning
• Search and Optimization: Local Search
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

A replacement action is a function $mathcal{L}$ that maps each graph $H$ to a collection of graphs of size at most $|V(H)|$. Given a graph class $mathcal{H}$, we consider a general family of graph modification problems, called $mathcal{L}$-Replacement to $mathcal{H}$, where the input is a graph $G$ and the question is whether it is possible to replace some induced subgraph $H_1$ of $G$ on at most $k$ vertices by a graph $H_2$ in $mathcal{L}(H_1)$ so that the resulting graph belongs to $mathcal{H}$. $mathcal{L}$-Replacement to $mathcal{H}$ can simulate many graph modification problems including vertex deletion, edge deletion/addition/edition/contraction, vertex identification, subgraph complementation, independent set deletion, (induced) matching deletion/contraction, etc. We present two algorithms. The first one solves $mathcal{L}$-Replacement to $mathcal{H}$ in time $2^{{ m poly}(k)}cdot |V(G)|^2$ for every minor-closed graph class $mathcal{H}$, where { m poly} is a polynomial whose degree depends on $mathcal{H}$, under a mild technical condition on $mathcal{L}$. This generalizes the results of Morelle, Sau, Stamoulis, and Thilikos [ICALP 2020, ICALP 2023] for the particular case of Vertex Deletion to $mathcal{H}$ within the same running time. Our second algorithm is an improvement of the first one when $mathcal{H}$ is the class of graphs embeddable in a surface of Euler genus at most $g$ and runs in time $2^{mathcal{O}(k^{9})}cdot |V(G)|^2$, where the $mathcal{O}(cdot)$ notation depends on $g$. To the best of our knowledge, these are the first parameterized algorithms with a reasonable parametric dependence for such a general family of graph modification problems to minor-closed classes.

Problem

• General graph modification problems to minor-closed classes
• Efficient algorithms for bounded-size replacement actions
• Parameterized solutions with polynomial parametric dependence

Innovation

• General graph replacement algorithm for minor-closed classes
• Polynomial-time parameterized algorithm for surface embeddable graphs
• Unified framework for diverse graph modification problems