# Surjective H-Colouring: New Hardness Results

## TL;DR

This paper investigates the computational complexity of the Surjective H-Colouring problem, establishing NP-completeness for a broad class of graphs and classifying complexity for all graphs up to four vertices.

## Contribution

It proves NP-completeness for Surjective H-Colouring on certain graphs and classifies the problem's complexity for all graphs with up to four vertices.

## Key findings

- NP-completeness for connected graphs with two non-adjacent self-loop vertices
- Complete classification of complexity for graphs with up to four vertices
- Extension of previous results on homomorphism problems

## Abstract

A homomorphism from a graph G to a graph H is a vertex mapping f from the vertex set of G to the vertex set of H such that there is an edge between vertices f(u) and f(v) of H whenever there is an edge between vertices u and v of G. The H-Colouring problem is to decide whether or not a graph G allows a homomorphism to a fixed graph H. We continue a study on a variant of this problem, namely the Surjective H-Colouring problem, which imposes the homomorphism to be vertex-surjective. We build upon previous results and show that this problem is NP-complete for every connected graph H that has exactly two vertices with a self-loop as long as these two vertices are not adjacent. As a result, we can classify the computational complexity of Surjective H-Colouring for every graph H on at most four vertices.