2 research outputs found
On the Complexity of Planar Covering of Small Graphs
The problem Cover(H) asks whether an input graph G covers a fixed graph H
(i.e., whether there exists a homomorphism G to 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. Kratochvil 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.Comment: Full version (including Appendix) of a paper from the conference WG
201
3-connected Reduction for Regular Graph Covers
A graph covers a graph if there exists a locally bijective
homomorphism from to . We deal with regular coverings in which this
homomorphism is prescribed by an action of a semiregular subgroup of
; so . In this paper, we study the
behaviour of regular graph covering with respect to 1-cuts and 2-cuts in .
We describe reductions which produce a series of graphs
such that is created from by replacing certain inclusion
minimal subgraphs with colored edges. The process ends with a primitive graph
which is either 3-connected, or a cycle, or . This reduction can be
viewed as a non-trivial modification of reductions of Mac Lane (1937),
Trachtenbrot (1958), Tutte (1966), Hopcroft and Tarjan (1973), Cuningham and
Edmonds (1980), Walsh (1982), and others. A novel feature of our approach is
that in each step all essential information about symmetries of are
preserved.
A regular covering projection induces regular covering
projections where is the -th quotient reduction of
. This property allows to construct all possible quotients of
from the possible quotients of . By applying this method to planar
graphs, we give a proof of Negami's Theorem (1988). Our structural results are
also used in subsequent papers for regular covering testing when is a
planar graph and for an inductive characterization of the automorphism groups
of planar graphs (see Babai (1973) as well).Comment: The journal version of the first part of arXiv:1402.377