On the complexity of planar covering of small graphs

Authors:
Ond$#345/ej Bí/lka;Jozef Jir$#225/sek;Pavel Klaví/k;Martin Tancer;Jan Volec
Affiliations:
Department of Applied Mathematics, Faculty of Mathematics and Physics, Charles University, Prague, Czech Republic;Department of Applied Mathematics, Faculty of Mathematics and Physics, Charles University, Prague, Czech Republic;Department of Applied Mathematics, Faculty of Mathematics and Physics, Charles University, Prague, Czech Republic;Department of Applied Mathematics, Faculty of Mathematics and Physics, Charles University, Prague, Czech Republic;Department of Applied Mathematics, Faculty of Mathematics and Physics, Charles University, Prague, Czech Republic
Venue:
WG'11 Proceedings of the 37th international conference on Graph-Theoretic Concepts in Computer Science
Year:
2011

Citing 10
Cited 0

Planar NAE3SAT is in P

ACM SIGACT News
The classification of coverings of processor networks

Journal of Parallel and Distributed Computing
On the complexity of H-coloring

Journal of Combinatorial Theory Series B
Scheduling algorithms for multihop radio networks

IEEE/ACM Transactions on Networking (TON)
Regular codes in regular graphs are difficult

Discrete Mathematics
Intersection graphs of segments

Journal of Combinatorial Theory Series B
Covering regular graphs

Journal of Combinatorial Theory Series B
Complexity of graph covering problems

Nordic Journal of Computing
List-Coloring Squares of Sparse Subcubic Graphs

SIAM Journal on Discrete Mathematics
On possible counterexamples to Negami's planar cover conjecture

Journal of Graph Theory

Quantified Score

Hi-index	0.00

Visualization

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.