Abstract
A graph G covers a graph H if there exists a locally bijective homomorphism from G to H. We deal with regular covers in which this locally bijective homomorphism is prescribed by an action of a subgroup of Aut(G). Regular covers have many applications in constructions and studies of big objects all over mathematics and computer science.
We study computational aspects of regular covers that have not been addressed before. The decision problem RegularCover asks for two given graphs G and H whether G regularly covers H. When |H| = 1, this problem becomes Cayley graph recognition for which the complexity is still unresolved. Another special case arises for |G| = |H| when it becomes the graph isomorphism problem. Therefore, we restrict ourselves to graph classes with polynomially solvable graph isomorphism.
Inspired by Negami, we apply the structural results used by Babai in the 1970’s to study automorphism groups of graphs. Our main result is an FPT algorithm solving RegularCover for planar input G in time \(\O^*(2^{e(H)/2})\) where e(H) denotes the number of the edges of H. In comparison, testing general graph covers is known to be NP-complete for planar inputs G even for small fixed graphs H such as K 4 or K 5. Most of our results also apply to general graphs, in particular the complete structural understanding of regular covers for 2-cuts.
For the full version of this paper, see arXiv:1402.3774. The first three authors are supported by the ESF Eurogiga project GraDR as GAČR GIG/11/E023, the fourth author by the ESF Eurogiga project GReGAS as the APVV project ESF-EC-0009-10. The first author is also supported by the project Kontakt LH12095. The second and the third authors are also supported by Charles University as GAUK 196213.
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Fiala, J., Klavík, P., Kratochvíl, J., Nedela, R. (2014). Algorithmic Aspects of Regular Graph Covers with Applications to Planar Graphs. In: Esparza, J., Fraigniaud, P., Husfeldt, T., Koutsoupias, E. (eds) Automata, Languages, and Programming. ICALP 2014. Lecture Notes in Computer Science, vol 8572. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-43948-7_41
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