Abstract
The notion of graph covers is a discretization of covering spaces introduced and deeply studied in topology. In discrete mathematics and theoretical computer science, they have attained a lot of attention from both the structural and complexity perspectives. Nonetheless, disconnected graphs were usually omitted from the considerations with the explanation that it is sufficient to understand coverings of the connected components of the target graph by components of the source one. However, different (but equivalent) versions of the definition of covers of connected graphs generalize to nonequivalent definitions of disconnected graphs. The aim of this paper is to summarize this issue and to compare three different approaches to covers of disconnected graphs: 1) locally bijective homomorphisms, 2) globally surjective locally bijective homomorphisms (which we call surjective covers), and 3) locally bijective homomorphisms which cover every vertex the same number of times (which we call equitable covers). The standpoint of our comparison is the complexity of deciding if an input graph covers a fixed target graph. We show that both surjective and equitable covers satisfy what certainly is a natural and welcome property: covering a disconnected graph is polynomial time decidable if such it is for every connected component of the graph, and it is NP-complete if it is NP-complete for at least one of its components. Despite of this, we argue that the third variant, equitable covers, is the right one, when considering covers of colored (multi)graphs. Moreover, the complexity of surjective and equitable covers differ from the fixed parameter complexity point of view. We conclude the paper by a complete characterization of the complexity of covering 2-vertex colored multigraphs with semi-edges. We present the results in the utmost generality and strength. In accord with the current trends we consider (multi)graphs with semi-edges, and, on the other hand, we aim at proving the NP-completeness results for simple input graphs.
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References
Abello, J., Fellows, M.R., Stillwell, J.C.: On the complexity and combinatorics of covering finite complexes. Aust. J. Combin. 4, 103–112 (1991)
Angluin, D.: Local and global properties in networks of processors. In: Proceedings of the 12th ACM Symposium on Theory of Computing, pp. 82–93 (1980)
Biggs, N.: Algebraic Graph Theory. Cambridge University Press, Cambridge (1974)
Bok, J., Fiala, J., Hliněný, P., Jedličková, N., Kratochvíl, J.: Computational complexity of covering two-vertex multigraphs with semi-edges. CoRR abs/2103.15214 (2021). https://arxiv.org/abs/2103.15214. To appear in proceedings of MFCS 2021
Chalopin, J., Métivier, Y., Zielonka, W.: Local computations in graphs: the case of cellular edge local computations. Fund. Inform. 74(1), 85–114 (2006)
Chaplick, S., Fiala, J., van ’t Hof, P., Paulusma, D., Tesař, M.: Locally constrained homomorphisms on graphs of bounded treewidth and bounded degree. In: Gąsieniec, L., Wolter, F. (eds.) FCT 2013. LNCS, vol. 8070, pp. 121–132. Springer, Heidelberg (2013). https://doi.org/10.1007/978-3-642-40164-0_14
Kratochvíl, J., Proskurowski, A., Telle, J.A.: Complexity of graph covering problems. In: Mayr, E.W., Schmidt, G., Tinhofer, G. (eds.) WG 1994. LNCS, vol. 903, pp. 93–105. Springer, Heidelberg (1995). https://doi.org/10.1007/3-540-59071-4_40
Kratochvíl, J., Proskurowski, A., Telle, J.A.: Covering directed multigraphs I. Colored directed multigraphs. In: Möhring, R.H. (ed.) WG. LNCS, vol. 1335, pp. 242–257. Springer, Heidelberg (1997)
Malnič, A., Nedela, R., Škoviera, M.: Lifting graph automorphisms by voltage assignments. Eur. J. Comb. 21(7), 927–947 (2000)
Matoušek, J., Nešetřil, J.: Invitation to Discrete Mathematics. Oxford University Press, Oxford (1998)
Mednykh, A.D., Nedela, R.: Harmonic Morphisms of Graphs: Part I: Graph Coverings, 1st edn. Vydavatelstvo Univerzity Mateja Bela v Banskej Bystrici (2015)
Acknowledgments
– Jan Bok and Nikola Jedličková: Supported by research grant GAČR 20-15576S of the Czech Science Foundation and by SVV–2020–260578. The authors were also partially supported by GAUK 1580119.
– Jiří Fiala and Jan Kratochvíl: Supported by research grant GAČR 20-15576S of the Czech Science Foundation.
– Michaela Seifrtová: Supported by research grant GAČR 19-17314J of the Czech Science Foundation.
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Bok, J., Fiala, J., Jedličková, N., Kratochvíl, J., Seifrtová, M. (2021). Computational Complexity of Covering Disconnected Multigraphs. In: Bampis, E., Pagourtzis, A. (eds) Fundamentals of Computation Theory. FCT 2021. Lecture Notes in Computer Science(), vol 12867. Springer, Cham. https://doi.org/10.1007/978-3-030-86593-1_6
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