LATIN 2014: LATIN 2014: Theoretical Informatics pp 526-537

# The Complexity of Homomorphisms of Signed Graphs and Signed Constraint Satisfaction

• Florent Foucaud
• Reza Naserasr
Conference paper
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.

## References

1. 1.
Rossi, F., van Beek, P., Walsh, T. (eds.): Handbook of Constraint Programming. Elsevier (2006)Google Scholar
2. 2.
Alon, N., Marshall, T.H.: Homomorphisms of edge-colored graphs and Coxeter groups. J. Algebr. Combin. 8(1), 5–13 (1998)
3. 3.
Brewster, R.C., Hell, P.: On homomorphisms to edge-coloured cycles. Electr. Notes Discrete Math. 5, 46–49 (2000)
4. 4.
Bulatov, A.A.: A dichotomy constraint on a three-element set. In: Proc. 43rd IEEE Symposium on Theory of Computing, pp. 649–658 (2002)Google Scholar
5. 5.
Charpentier, C., Naserasr, R., Sopena, E.: Analogue of Jeager-Zhang conjecture for signed bipartite graphs (manuscript)Google Scholar
6. 6.
Demaine, E., Hajiaghayi, M., Kawarabayashi, K.-I.: Decomposition, Approximation, and Coloring of Odd-Minor-Free Graphs. In: Proc. SODA 2010, pp. 329–344 (2010)Google Scholar
7. 7.
Feder, T., Vardi, M.Y.: The Computational structure of monotone monadic SNP and constraint catisfaction: a study through datalog and group theory. SIAM J. Comput. 28(1), 57–104 (1998)
8. 8.
Geelen, J., Gerards, B., Reed, B., Seymour, P., Vetta, A.: On the odd-minor variant of Hadwigers conjecture. J. Combin. Theor. Series B 99, 20–29 (2009)
9. 9.
Guenin, B.: Graphs without odd-K5 minors are 4-colourable. Talk at Oberwolfach Seminar on Graph Theory (January 2005)Google Scholar
10. 10.
Guenin, B.: Packing odd circuit covers: a conjecture (2005) (unpublished manuscript)Google Scholar
11. 11.
Hell, P., Nešetřil, J.: On the complexity of H-coloring. J. Combin. Theor. Series B 48(1), 92–110 (1990)
12. 12.
Hell, P., Nešetřil, J., Zhu, X.: Complexity of tree homomorphisms. Discrete Appl. Math. 70, 23–36 (1996)
13. 13.
Moret, B.M.E.: The Theory of Computation, ch. 7, Problem 7.1, Part 2. Addison-Wesley (1998)Google Scholar
14. 14.
Naserasr, R., Rollová, E., Sopena, E.: Homomorphisms of signed graphs (submitted manuscript)Google Scholar
15. 15.
Zaslavsky, T.: Signed graphs. Discrete Appl. Math. 4(1), 47–74 (1982)

## 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