Optimal Path Partitions in Subcubic and Almost-subcubic Graphs

2026-02-13

Technology Category

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

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

We consider the problem of partitioning the edges of a graph into as few paths as possible. This is a~subject of the classic conjecture of Gallai and a recurring topic in combinatorics. Regarding the complexity of partitioning a graph optimally, Peroch\'e [Discret. Appl. Math., 1984] proved that it is NP-hard already on graphs of maximum degree four, even when we only ask if two paths suffice. We show that the problem is solvable in polynomial time on subcubic graphs and then we present an efficient algorithm for ``almost-subcubic''graphs. Precisely, we prove that the problem is fixed-parameter tractable when parameterized by the edge-deletion distance to a subcubic graph. To this end, we reduce the task to model checking in first-order logic extended by disjoint-paths predicates ($\mathsf{FO}\text{+}\mathsf{DP}$) and then we employ the recent tractability result by Schirrmacher, Siebertz, Stamoulis, Thilikos, and Vigny [LICS 2024].

Problem

• path partition
• subcubic graph
• edge partition
• Gallai conjecture
• graph decomposition

Innovation

• subcubic graphs
• fixed-parameter tractability
• edge partition
• first-order logic with disjoint paths
• polynomial-time algorithm