Finding an induced subdivision of a digraph

TL;DR

This paper investigates the computational complexity of detecting induced subdivisions of a fixed digraph within larger graphs, identifying both polynomial-time solvable cases and NP-complete instances.

Contribution

It provides a systematic analysis of the problem's complexity depending on the target digraph and graph restrictions, including new polynomial and NP-complete cases.

Findings

Identifies polynomial-time solvable instances.

Establishes NP-completeness for several cases.

Analyzes complexity based on digraph and graph restrictions.

Abstract

We consider the following problem for oriented graphs and digraphs: Given an oriented graph (digraph) $G$, does it contain an induced subdivision of a prescribed digraph $D$? The complexity of this problem depends on $D$ and on whether $G$ must be an oriented graph or is allowed to contain 2-cycles. We give a number of examples of polynomial instances as well as several NP-completeness proofs.