Abstract
The problem Cover(H) asks whether an input graph G covers a fixed graph H (i.e., whether there exists a homomorphism G → H which locally preserves the structure of the graphs). Complexity of this problem has been intensively studied. In this paper, we consider the problem PlanarCover(H) which restricts the input graph G to be planar.
PlanarCover(H) is polynomially solvable if Cover(H) belongs to P, and it is even trivially solvable if H has no planar cover. Thus the interesting cases are when H admits a planar cover, but Cover(H) is NP-complete. This also relates the problem to the long-standing Negami Conjecture which aims to describe all graphs having a planar cover. Kratochvíl asked whether there are non-trivial graphs for which Cover(H) is NP-complete but Planarcover(H) belongs to P.
We examine the first nontrivial cases of graphs H for which Cover(H) is NP-complete and which admit a planar cover. We prove NP-completeness of Planarcover(H) in these cases.
The initial research was supported by DIMACS/DIMATIA REU program (grant number 0648985). The third and the fourth author were supported by Charles University as GAUK 95710. The fourth author is also affiliated with Institute for Theoretical Computer Science (supported by project 1M0545 of The Ministry of Education of the Czech Republic).
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
Abello, J., Fellows, M.R., Stillwell, J.C.: On the complexity and combinatorics of covering finite complexes. Australian Journal of Combinatorics 4, 103–112 (1991)
Bodlaender, H.L.: The classification of coverings of processor networks. Journal of Parallel and Distributed Computing 6(1), 166–182 (1989)
Dvořák, Z., Škrekovski, R., Tancer, M.: List-coloring squares of sparse subcubic graphs. SIAM J. Discrete Math. 22(1), 139–159 (2008)
Ehrlich, G., Even, S., Tarjan, R.E.: Intersection graphs of curves in the plane. Journal of Combinatorial Theory, Series B 21(1), 8–20 (1976)
Fiala, J.: Note on the computational complexity of covering regular graphs. In: 9th Annual Conference of Doctoral Students, WDS 2000, pp. 89–90. Matfyzpress (2000)
Hell, P., Nešetřil, J.: On the complexity of H-coloring. J. Combin. Theory Ser. B 48(1), 92–110 (1990)
Hliněný, P., Thomas, R.: On possible counterexamples to Negami’s planar cover conjecture. J. Graph Theory 46(3), 183–206 (2004)
Janczewski, R., Kosowski, A., Malafiejski, M.: The complexity of the l(p,q)-labeling problem for bipartite planar graphs of small degree. Discrete Mathematics 309(10), 3270–3279 (2009)
Kára, J., Kratochvíl, J., Wood, D.R.: On the complexity of the balanced vertex ordering problem. Discrete Mathematics & Theoretical Computer Science 9(1), 193–202 (2007)
Kratochvíl, J., Matoušek, J.: Intersection graphs of segments. Journal of Combinatorial Theory, Series B 62(2), 289–315 (1994)
Kratochvíl, J., Proskurowski, A., Telle, J.A.: Covering regular graphs. J. Comb. Theory Ser. B 71(1), 1–16 (1997)
Kratochvíl, J., Proskurowski, A., Telle, J.A.: Complexity of graph covering problems. Nordic J. of Computing 5(3), 173–195 (1998)
Kratochvíl, J.: Regular codes in regular graphs are difficult. Discrete Math. 133(1-3), 191–205 (1994)
Moret, B.M.E.: Planar NAE3SAT is in P. SIGACT News 19, 51–54 (1988)
Negami, S.: The spherical genus and virtually planar graphs. Discrete Math. 70(2), 159–168 (1988)
Ramanathan, S., Lloyd, E.: Scheduling algorithms for multihop radio networks. IEEE/ACM Transactions on Networking 1(2), 166–177 (1993)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2011 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Bílka, O., Jirásek, J., Klavík, P., Tancer, M., Volec, J. (2011). On the Complexity of Planar Covering of Small Graphs. In: Kolman, P., Kratochvíl, J. (eds) Graph-Theoretic Concepts in Computer Science. WG 2011. Lecture Notes in Computer Science, vol 6986. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-25870-1_9
Download citation
DOI: https://doi.org/10.1007/978-3-642-25870-1_9
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-25869-5
Online ISBN: 978-3-642-25870-1
eBook Packages: Computer ScienceComputer Science (R0)