Optimal Path Partitions in Subcubic and Almost-subcubic Graphs

TL;DR

This paper investigates the complexity of partitioning graph edges into the fewest paths, showing polynomial-time solutions for subcubic graphs and fixed-parameter tractability for almost-subcubic graphs using logical model checking.

Contribution

It proves polynomial-time solvability for subcubic graphs and fixed-parameter tractability for almost-subcubic graphs based on edge-deletion distance.

Findings

Polynomial-time algorithm for subcubic graphs

Fixed-parameter tractability for almost-subcubic graphs

Reduction to model checking in extended first-order logic

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].