# The Hardness of Embedding Grids and Walls

## TL;DR

This paper investigates the computational complexity of embedding problems for specific graph classes, proving $W[1]$-completeness for grids and walls, thus advancing understanding of the dichotomy conjecture.

## Contribution

It establishes the $W[1]$-completeness of the embedding problem for grid and wall graph classes, providing key evidence towards the conjecture.

## Key findings

- Embedding problem is $W[1]$-complete for grids.
- Embedding problem is $W[1]$-complete for walls.
- Supports the dichotomy conjecture for bounded tree width classes.

## Abstract

The dichotomy conjecture for the parameterized embedding problem states that the problem of deciding whether a given graph $G$ from some class $K$ of "pattern graphs" can be embedded into a given graph $H$ (that is, is isomorphic to a subgraph of $H$) is fixed-parameter tractable if $K$ is a class of graphs of bounded tree width and $W[1]$-complete otherwise. Towards this conjecture, we prove that the embedding problem is $W[1]$-complete if $K$ is the class of all grids or the class of all walls.