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The Complexity of Homomorphisms of Signed Graphs and Signed Constraint Satisfaction

  • Florent Foucaud
  • Reza Naserasr
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8392)

Abstract

A signed graph (G, Σ) is an undirected graph G together with an assignment of signs (positive or negative) to all its edges, where Σ denotes the set of negative edges. Two signatures are said to be equivalent if one can be obtained from the other by a sequence of resignings (i.e. switching the sign of all edges incident to a given vertex). Extending the notion of usual graph homomorphisms, homomorphisms of signed graphs were introduced, and have lead to some extensions and strengthenings in the theory of graph colorings and homomorphisms. We study the complexity of deciding whether a given signed graph admits a homomorphism to a fixed target signed graph [H,Σ], i.e. the (H,Σ)-Coloring problem. We prove a dichotomy result for the class of all (C k ,Σ)-Coloring problems (where C k is a cycle of length k ≥ 3): (C k ,Σ)-Coloring is NP-complete, unless both k and the size of Σ are even. We conjecture that this dichotomy can be extended to all signed graphs in a natural way. We also introduce the more general concept of signed constraint satisfaction problems and show that a dichotomy for such problems is equivalent to the statement of the Feder-Vardi Dichotomy Conjecture.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  • Florent Foucaud
    • 1
    • 2
    • 3
  • Reza Naserasr
    • 4
  1. 1.Universitat Politècnica de CatalunyaBarcelonaSpain
  2. 2.University of JohannesburgAuckland ParkSouth Africa
  3. 3.PSL, LAMSADE - CNRS UMR 7243Université Paris-DauphineFrance
  4. 4.CNRS, LRI - CNRS UMR 8623Université Paris-Sud 11OrsayFrance

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