Finding vertex-surjective graph homomorphisms

TL;DR

This paper investigates the computational complexity of finding vertex-surjective homomorphisms between graphs, revealing polynomial-time solvability in some cases and NP-completeness in others, depending on graph classes.

Contribution

It characterizes the complexity of the Surjective Homomorphism problem for various graph classes and identifies fixed-parameter tractability results.

Findings

Polynomial-time solvable when H is the class of paths.

NP-complete when G is the class of paths.

Fixed-parameter tractable for certain tree and vertex cover classes.

Abstract

The Surjective Homomorphism problem is to test whether a given graph G called the guest graph allows a vertex-surjective homomorphism to some other given graph H called the host graph. The bijective and injective homomorphism problems can be formulated in terms of spanning subgraphs and subgraphs, and as such their computational complexity has been extensively studied. What about the surjective variant? Because this problem is NP-complete in general, we restrict the guest and the host graph to belong to graph classes G and H, respectively. We determine to what extent a certain choice of G and H influences its computational complexity. We observe that the problem is polynomial-time solvable if H is the class of paths, whereas it is NP-complete if G is the class of paths. Moreover, we show that the problem is even NP-complete on many other elementary graph classes, namely linear forests, unions of complete graphs, cographs, proper interval graphs, split graphs and trees of pathwidth at most 2. In contrast, we prove that the problem is fixed-parameter tractable in k if G is the class of trees and H is the class of trees with at most k leaves, or if G and H are equal to the class of graphs with vertex cover number at most k.