A note on the structure of locally finite planar quasi-transitive graphs

TL;DR

This paper extends the structure theory of planar quasi-transitive graphs by establishing a canonical tree-decomposition for 3-connected locally finite cases, generalizing Droms' theorem and providing new insights into their embeddings.

Contribution

It provides a general structure theorem for 3-connected locally finite planar quasi-transitive graphs, including a canonical tree-decomposition aligned with cycle-separations.

Findings

Every such graph admits a canonical tree-decomposition.

Parts of the decomposition have vertex-accumulation free embeddings.

Alternative proof that these graphs have decompositions with 1-ended or finite planar parts.

Abstract

In an early work from 1896, Maschke established the complete list of all finite planar Cayley graphs. This result initiated a long line of research over the next century, aiming at characterizing in a similar way all planar infinite Cayley graphs. Droms (2006) proved a structure theorem for finitely generated planar groups, i.e., finitely generated groups admitting a planar Cayley graph, in terms of Bass-Serre decompositions. As a byproduct of his structure theorem, Droms proved that such groups are finitely presented. More recently, Hamann (2018) gave a graph theoretical proof that every planar quasi-transitive graph $G$ admits a generating $\mathrm{Aut}(G)$-invariant set of closed walks with only finitely many orbits, and showed that a consequence is an alternative proof of Droms' result. Based on the work of Hamann, we show in this note that we can also obtain a general structure theorem for $3$-connected locally finite planar quasi-transitive graphs, namely that every such graph admits a canonical tree-decomposition whose edge-separations correspond to cycle-separations in the (unique) embedding of $G$, and in which every part admits a vertex-accumulation free embedding. This result can be seen as a version of Droms' structure theorem for quasi-transitive planar graphs. As a corollary, we obtain an alternative proof of a result of Hamann, Lehner, Miraftab and R\"uhmann (2022) that every locally finite quasi-transitive planar graph admits a canonical tree-decomposition, whose parts are either $1$-ended or finite planar graphs.