# The Almost-Disjoint 2-Path Decomposition Problem

## TL;DR

This paper studies the problem of decomposing directed graphs into length-2 paths with limited vertex overlap, proving NP-hardness and identifying special graph classes where the problem is tractable.

## Contribution

It establishes NP-hardness of the problem, characterizes graph classes where it reduces to the Stable-set problem, and provides a dynamic programming solution for series-parallel digraphs.

## Key findings

- NP-hardness proven via reduction from 3-SAT
- Characterization of graph classes reducible to Stable-set problem
- Dynamic programming approach for series-parallel digraphs

## Abstract

We consider the problem of decomposing a given (di)graph into paths of length 2 with the additional restriction that no two such paths may have more than one vertex in common. We establish its NP-hardness by a reduction from 3-SAT, characterize (di)graph classes for which the problem can be be reduced to the Stable-set problem on claw-free graphs and describe a dynamic program for solving it for series-parallel digraphs.