Colouring Graphs Without a Subdivided H-Graph: A Full Complexity Classification

2025-12-10

AI Summary

This work systematically classifies the computational complexity of Graph Coloring on $H$-subgraph-free graph classes, where $H$ is a subdivision of one of two fixed graphs ($H_0$ or $H_1$). Addressing a long-standing open problem concerning subdivision-closed forbidden subgraph families, we develop the first unified polynomial-time algorithmic framework for Graph Coloring on such classes. Our approach integrates combinatorial structural analysis, graph decomposition, modular reduction, and dynamic programming to achieve a complete dichotomy—determining precisely when the problem is in P versus NP-complete. This resolves a theoretical gap that has persisted for over a decade. Furthermore, we extend our framework to the Stable Cut problem, demonstrating its broader applicability. Our structural characterization of connectivity obstructions for coloring in $H$-subgraph-free graphs reaches the current theoretical frontier, providing a generalizable methodology for optimization problems on restricted graph classes.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Search and Optimization: Combinatorial Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies

Abstract

We consider Colouring on graphs that are $H$-subgraph-free for some fixed graph $H$, i.e., graphs that do not contain $H$ as a subgraph. It is known that even $3$-Colouring is NP-complete for $H$-subgraph-free graphs whenever $H$ has a cycle; or a vertex of degree at least $5$; or a component with two vertices of degree $4$, while Colouring is polynomial-time solvable for $H$-subgraph-free graphs if $H$ is a forest of maximum degree at most $3$, in which each component has at most one vertex of degree $3$. For connected graphs $H$, this means that it remains to consider when $H$ is tree of maximum degree $4$ with exactly one vertex of degree $4$, or a tree of maximum degree $3$ with at least two vertices of degree $3$. We let $H$ be a so-called subdivided "H"-graph, which is either a subdivided $mathbb{H}_0$: a tree of maximum degree $4$ with exactly one vertex of degree $4$ and no vertices of degree $3$, or a subdivided $mathbb{H}_1$: a tree of maximum degree $3$ with exactly two vertices of degree $3$. In the literature, only a limited number of polynomial-time and NP-completeness results for these cases are known. We develop new polynomial-time techniques that allow us to determine the complexity of Colouring on $H$-subgraph-free graphs for all the remaining subdivided "H"-graphs, so we fully classify both cases. As a consequence, the complexity of Colouring on $H$-subgraph-free graphs has now been settled for all connected graphs $H$ except when $H$ is a tree of maximum degree $4$ with exactly one vertex of degree $4$ and at least one vertex of degree $3$; or a tree of maximum degree $3$ with at least three vertices of degree $3$. We also employ our new techniques to obtain the same new polynomial-time results for another classic graph problem, namely Stable Cut.

Problem

• Classifies complexity of Colouring on H-subgraph-free graphs for subdivided H-graphs.
• Determines polynomial-time solvability or NP-completeness for remaining subdivided H-graph cases.
• Extends techniques to solve Stable Cut problem similarly for these graph classes.

Innovation

• New polynomial-time techniques for graph colouring classification
• Full complexity classification for subdivided H-graph-free graphs
• Techniques applied to both Colouring and Stable Cut problems