Abstract
A homomorphism from a graph G to a graph H is locally bijective, surjective, or injective if its restriction to the neighborhood of every vertex of G is bijective, surjective, or injective, respectively. We prove that the problems of testing whether a given graph G allows a homomorphism to a given graph H that is locally bijective, surjective, or injective, respectively, are NP-complete, even when G has pathwidth at most 5, 4 or 2, respectively, or when both G and H have maximum degree 3. We complement these hardness results by showing that the three problems are polynomial-time solvable if G has bounded treewidth and in addition G or H has bounded maximum degree.
This paper is supported by the Natural Sciences Engineering Research Council of Canada (NSERC), the Research Council of Norway (197548/F20), EPSRC (EP/G043434/1) and the Royal Society (JP100692).
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References
Angluin, D.: Local and global properties in networks of processors. In: Proc. STOC 1980, pp. 82–93 (1980)
Angluin, D., Gardiner, A.: Finite common coverings of pairs of regular graphs. J. Comb. Theory Ser. B 30, 184–187 (1981)
Biggs, N.: Algebraic Graph Theory. Cambridge University Press (1974)
Biggs, N.: Constructing 5-arc transitive cubic graphs. J. London Math. Society II 26, 193–200 (1982)
Bodlaender, H.L.: The classification of coverings of processor networks. J. Par. Distrib. Comp. 6, 166–182 (1989)
Bodlaender, H.L.: A linear-time algorithm for finding tree-decompositions of small treewidth. SIAM J. Comp. 25(6), 1305–1317 (1996)
Chalopin, J., Métivier, Y., Zielonka, W.: Election, naming and cellular edge local computations. In: Ehrig, H., Engels, G., Parisi-Presicce, F., Rozenberg, G. (eds.) ICGT 2004. LNCS, vol. 3256, pp. 242–256. Springer, Heidelberg (2004)
Dalmau, V., Kolaitis, P.G., Vardi, M.Y.: Constraint satisfaction, bounded treewidth, and finite-variable logics. In: Van Hentenryck, P. (ed.) CP 2002. LNCS, vol. 2470, pp. 310–326. Springer, Heidelberg (2002)
Everett, M.G., Borgatti, S.: Role coloring a graph. Mathematical Social Sciences 21, 183–188 (1991)
Fiala, J., Kratochvíl, J.: Locally constrained graph homomorphisms – Structure, complexity, and applications. Comp. Sci. Review 2, 97–111 (2008)
Fiala, J., Kratochvíl, J.: Partial covers of graphs. Disc. Math. Graph Theory 22, 89–99 (2002)
Fiala, J., Kratochvíl, J., Kloks, T.: Fixed-parameter complexity of λ-labelings. Discr. Appl. Math. 113, 59–72 (2001)
Fiala, J., Paulusma, D.: A complete complexity classification of the role assignment problem. Theor. Comp. Sci. 349, 67–81 (2005)
Fiala, J., Paulusma, D.: Comparing universal covers in polynomial time. Theory Comp. Syst. 46, 620–635 (2010)
Galluccio, A., Hell, P., Nešetřil, J.: The complexity of H-colouring of bounded degree graphs. Discr. Math. 222, 101–109 (2000)
Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-completeness. W. H. Freeman & Co., New York (1979)
Grohe, M.: The complexity of homomorphism and constraint satisfaction problems seen from the other side. J. ACM 54 (2007)
Gurski, F., Wanke, E.: The tree-width of clique-width bounded graphs without K n,n . In: Brandes, U., Wagner, D. (eds.) WG 2000. LNCS, vol. 1928, pp. 196–205. Springer, Heidelberg (2000)
Hell, P., Nešetřil, J.: On the complexity of H-colouring. J. Comb. Theory Ser. B 48, 92–110 (1990)
Hell, P., Nešetřil, J.: Graphs and Homomorphisms. Oxford University Press (2004)
Heggernes, P., van ’t Hof, P., Paulusma, D.: Computing role assignments of proper interval graphs in polynomial time. J. Discr. Alg. 14, 173–188 (2012)
Kloks, T.: Treewidth, Computations and Approximations. LNCS, vol. 842. Springer (1994)
Kratochvíl, J., Křivánek, M.: On the computational complexity of codes in graphs. In: Koubek, V., Janiga, L., Chytil, M.P. (eds.) MFCS 1988. LNCS, vol. 324, pp. 396–404. Springer, Heidelberg (1988)
Kratochvíl, J., Proskurowski, A., Telle, J.A.: Covering regular graphs. J. Comb. Theory Ser. B 71, 1–16 (1997)
Kristiansen, P., Telle, J.A.: Generalized H-coloring of graphs. In: Lee, D.T., Teng, S.-H. (eds.) ISAAC 2000. LNCS, vol. 1969, pp. 456–466. Springer, Heidelberg (2000)
Massey, W.S.: Algebraic Topology: An Introduction. Harcourt, Brace and World (1967)
Nešetřil, J.: Homomorphisms of derivative graphs. Discr. Math. 1, 257–268 (1971)
Pekeč, A., Roberts, F.S.: The role assignment model nearly fits most social networks. Mathematical Social Sciences 41, 275–293 (2001)
Roberts, F.S., Sheng, L.: How hard is it to determine if a graph has a 2-role assignment? Networks 37, 67–73 (2001)
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Chaplick, S., Fiala, J., van ’t Hof, P., Paulusma, D., Tesař, M. (2013). Locally Constrained Homomorphisms on Graphs of Bounded Treewidth and Bounded Degree. In: Gąsieniec, L., Wolter, F. (eds) Fundamentals of Computation Theory. FCT 2013. Lecture Notes in Computer Science, vol 8070. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-40164-0_14
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