On the Complexity of Planar Covering of Small Graphs

TL;DR

This paper investigates the computational complexity of the PlanarCover problem for graphs, showing NP-completeness in certain cases and relating it to the Negami Conjecture, thus advancing understanding of planar graph coverings.

Contribution

It proves NP-completeness of PlanarCover(H) for specific graphs H where Cover(H) is NP-complete and H admits a planar cover, addressing a key open question.

Findings

NP-completeness of PlanarCover(H) for certain graphs H

Relation to the Negami Conjecture clarified

Progress in understanding planar graph coverings

Abstract

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.