# Subgraph Isomorphism on Graph Classes that Exclude a Substructure

## TL;DR

This paper explores the complexity of Subgraph Isomorphism on graph classes defined by forbidden minors, establishing a near dichotomy and identifying cases that are fixed-parameter tractable or NP-complete.

## Contribution

It provides a near-complete complexity classification for Subgraph Isomorphism on minor-closed graph classes, focusing on the role of the forbidden minor and introducing new fixed-parameter tractability results.

## Key findings

- Fixed-parameter tractability for certain minor-closed classes
- NP-completeness for generalized cases close to known tractable ones
- Identification of P5 as the key unsettled case

## Abstract

We study Subgraph Isomorphism on graph classes defined by a fixed forbidden graph. Although there are several ways for forbidding a graph, we observe that it is reasonable to focus on the minor relation since other well-known relations lead to either trivial or equivalent problems. When the forbidden minor is connected, we present a near dichotomy of the complexity of Subgraph Isomorphism with respect to the forbidden minor, where the only unsettled case is $P_{5}$, the path of five vertices. We then also consider the general case of possibly disconnected forbidden minors. We show fixed-parameter tractable cases and randomized XP-time solvable cases parameterized by the size of the forbidden minor $H$. We also show that by slightly generalizing the tractable cases, the problem becomes NP-complete. All unsettle cases are equivalent to $P_{5}$ or the disjoint union of two $P_{5}$'s. As a byproduct, we show that Subgraph Isomorphism is fixed-parameter tractable parameterized by vertex integrity. Using similar techniques, we also observe that Subgraph Isomorphism is fixed-parameter tractable parameterized by neighborhood diversity.