A note on the structure of locally finite planar quasi-transitive graphs (2412.20300v1)

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.

• The paper demonstrates a canonical tree-decomposition for every locally finite, 3-connected quasi-transitive planar graph.
• It employs cycle-separation techniques linked to vertex-accumulation free embeddings to mirror classical planar graph theorems.
• Findings offer practical insights for efficient graph algorithms and combinatorial optimization in complex networks.

Structure of Quasi-Transitive Planar Graphs

The paper presented in this paper examines the intrinsic structure of locally finite planar quasi-transitive graphs, contributing to our understanding of the broader class of planar graphs. This exploration is rooted in the historical context established by Maschke's classification of finite planar Cayley graphs in 1896, which set the stage for subsequent studies seeking to characterize infinite planar Cayley graphs. Over time, significant advancements have been made in understanding such infinite graphs through various decompositions, such as those proving that finitely generated planar groups are finitely presented (Droms, 1985).

The core of this research investigates the potential of extending known decompositions to quasi-transitive planar graphs, specifically focusing on the construction of a canonical tree-decomposition for these graphs. The paper effectively demonstrates that every locally finite quasi-transitive $3$-connected planar graph admits a canonical tree-decomposition. This decomposition is characterized by edge-separations corresponding to cycle-separations in the graph’s embedding, with each part admitting an embedding that is vertex-accumulation free. Such a decomposition mirrors Droms' structure theorem for quasi-transitive planar graphs and provides an alternative proof that all locally finite quasi-transitive planar graphs have canonical tree-decompositions, whereby each component is either finite or $1$-ended.

The implications of these findings are both theoretical and practical. On a theoretical plane, they extend the understanding of the structural properties of infinite graphs, particularly in verifying that planar graphs within the quasi-transitive class can achieve similar decompositional outcomes as transitive graphs, previously linked to Coyley graphs. Practically, these results lend themselves to applications around efficient graph algorithms, given that canonical decompositions are foundational in designing efficient data structuring and searching algorithms on graphs.

Future research should investigate the scope of these decompositions in other minor-excluding graph classes or explore their impact on the development of algorithms in combinatorial optimizations. Additionally, examining the applicability of these structured decompositions in more complex networks, such as those found in computational fields or physical systems modeling, could prove beneficial. The potential expansion of these methodologies could significantly innovate the realms dealing with interaction and connectivity modeling, effectively bridging theoretical graph studies with applied computational technologies.