Acta Informatica

, Volume 49, Issue 6, pp 381–394 | Cite as

Finding vertex-surjective graph homomorphisms

  • Petr A. Golovach
  • Bernard Lidický
  • Barnaby Martin
  • Daniël PaulusmaEmail author
Original Article


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 \({{\mathcal G}}\) and \({{\mathcal H}}\), respectively. We determine to what extent a certain choice of \({{\mathcal G}}\) and \({{\mathcal H}}\) influences its computational complexity. We observe that the problem is polynomial-time solvable if \({{\mathcal H}}\) is the class of paths, whereas it is NP-complete if \({{\mathcal G}}\) 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 \({{\mathcal G}}\) is the class of trees and \({{\mathcal H}}\) is the class of trees with at most k leaves, or if \({{\mathcal G}}\) and \({{\mathcal H}}\) are equal to the class of graphs with vertex cover number at most k.


Complete Graph Vertex Cover Hamiltonian Path Tree Decomposition Graph Class 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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Copyright information

© Springer-Verlag 2012

Authors and Affiliations

  • Petr A. Golovach
    • 1
  • Bernard Lidický
    • 2
    • 3
  • Barnaby Martin
    • 1
  • Daniël Paulusma
    • 1
    Email author
  1. 1.Science Laboratories, School of Engineering and Computing SciencesDurham UniversityDurhamUK
  2. 2.Faculty of Mathematics and PhysicsCharles UniversityPragueCzech Republic
  3. 3.Department of MathematicsUniversity of IllinoisUrbanaUSA

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