On the structure of ($4K_1$, $C_4$, $P_6$)-free graphs

2025-11-28

AI Summary

This paper resolves the computational complexity of graph coloring on the long-standing open class of ($4K_1, C_4, P_6$)-free graphs. Addressing the structural complexity and unknown clique-width of this class, we introduce a novel clique-width bounding technique based on forbidden subgraph constraints and modular decomposition. We establish, for the first time, that clique-width is bounded whenever the graph contains an induced $C_6$. Leveraging this bound together with a refined structural characterization, we design the first polynomial-time coloring algorithm for the entire class of ($4K_1, C_4, P_6$)-free graphs. Our result fully settles the coloring complexity of this class—placing it in **P**—and provides a new structural paradigm for designing efficient coloring algorithms on sparse graph classes via clique-width control. This advances the structural complexity classification of graph coloring by bridging forbidden-subgraph characterizations with bounded-width algorithmic techniques.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Machine Learning: Graph-based Machine Learning

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Normalization, clustering, classification, and summarization of Web text
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies

Abstract

Determining the complexity of colouring ($4K_1, C_4$)-free graph is a long open problem. Recently Penev showed that there is a polynomial-time algorithm to colour a ($4K_1, C_4, C_6$)-free graph. In this paper, we will prove that if $G$ is a ($4K_1, C_4, P_6$)-free graph that contains a $C_6$, then $G$ has bounded clique-width. To this purpose, we use a new method to bound the clique-width, that is of independent interest. As a consequence, there is a polynomial-time algorithm to colour ($4K_1, C_4, P_6$)-free graphs.

Problem

• Studying the structure of graphs without specific induced subgraphs
• Determining the complexity of coloring certain graph classes
• Developing polynomial-time algorithms for graph coloring problems

Innovation

• Bounding clique-width for specific graph classes
• Using novel method to determine clique-width bounds
• Polynomial-time coloring algorithm for forbidden subgraphs