3-connected Reduction for Regular Graph Covers

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 semiregular subgroup $\Gamma$ of $\mathrm{Aut}(G)$; so $H \cong G / \Gamma$. In this paper, we study behaviour of regular graph covering with respect to 1-cuts and 2-cuts in $G$. An atom is an inclusion-minimal subgraph which is essentially 3-connected and cannot be further simplified. We use reductions which produce a series of graphs $G = G_0,\dots,G_r$ such that $G_{i+1}$ is created from $G_i$ by replacing all atoms with colored edges. The primitive graph $G_r$ contains no atoms and it is either essentially 3-connected, or it is essentially a cycle, or $K_2$. Our aim is to revert the reduction and construct a quotient $H_0=G_0/\Gamma_0$ from $H_r=G_r/\Gamma_r$. This process is called an expansion and it creates a series $H_r,H_{r-1},\dots,H_0$ with an extension series of semiregular subgroups $\Gamma_r,\Gamma_{r-1},\dots,\Gamma_0$. Inspired by Negami (1988), we show that an atom might have three types of quotients: the unique edge-quotient, the unique loop-quotient, and possibly several half-quotients. Our main result states that all graphs $H_i$ which can be created from $H_{i+1}$ by replacing edges with the edge-quotiens, loops with the loop-quotients and half-edges with some choices of half-quotients. Applying the reductions and expansions to planar graphs, we obtain two structural results. First, we characterize the automorphism groups of planar graphs, similarly to Babai (1975). Second, we give a direct proof of Negami's Theorem (1988) by describing all possible quotients of planar graphs. Also, this paper develops a theoretical background for an investigation of the complexity of testing whether a given input graph $G$ regularly covers another input graph $H$.