Finding a subdivision of a prescribed digraph of order 4

TL;DR

This paper classifies the computational complexity of detecting subdivisions of 4-vertex digraphs, showing most cases are either polynomial-time solvable or NP-complete, advancing understanding of digraph subdivision problems.

Contribution

It provides a near-complete classification of the algorithmic complexity for finding subdivisions of 4-vertex digraphs, supporting open conjectures.

Findings

Most 4-vertex digraph subdivision problems are classified as polynomial or NP-complete.

All NP-hardness results are derived from reductions of the 2-linkage problem.

Some polynomial cases involve complex algorithms.

Abstract

The problem of when a given digraph contains a subdivision of a fixed digraph $F$ is considered. Bang-Jensen et al. laid out foundations for approaching this problem from the algorithmic point of view. In this paper we give further support to several open conjectures and speculations about algorithmic complexity of finding $F$-subdivisions. In particular, up to 5 exceptions, we completely classify for which 4-vertex digraphs $F$, the $F$-subdivision problem is polynomial-time solvable and for which it is NP-complete. While all NP-hardness proofs are made by reduction from some version of the 2-linkage problem in digraphs, some of the polynomial-time solvable cases involve relatively complicated algorithms.