[Paper Review] The planar Cayley graphs are effectively enumerable

Agelos Georgakopoulos, Matthias Hamann|arXiv (Cornell University)|Jun 10, 2015

semigroups and automata theory26 references3 citations

TL;DR

This paper establishes that planar Cayley graphs—Cayley graphs realizable in the plane without edge crossings—can be effectively enumerated by introducing a new class of group presentations called planar presentations. These presentations are algorithmically recognizable, enabling a systematic, computable enumeration of all such graphs and resolving an open question on the effective enumerability of planar groups.

ABSTRACT

We show that a group admits a planar, finitely generated Cayley graph if and only if it admits a special kind of group presentation we introduce, called a planar presentation. Planar presentations can be recognised algorithmically. As a consequence, we obtain an effective enumeration of the planar Cayley graphs, yielding in particular an affirmative answer to a question of Droms et al. asking whether the planar groups can be effectively enumerated.

Motivation & Objective

• To determine whether planar Cayley graphs can be effectively enumerated, addressing an open question by Droms et al.
• To characterize groups admitting planar, finitely generated Cayley graphs through a new class of group presentations.
• To introduce and formalize the concept of planar presentations as a tool for algorithmic recognition of planar Cayley graphs.
• To establish a constructive, algorithmic link between group presentations and planar graph realizations.

Proposed method

• Introduce the notion of a planar presentation—a specific type of group presentation whose associated Cayley graph is planar and finitely generated.
• Prove that a group admits a planar, finitely generated Cayley graph if and only if it admits a planar presentation.
• Demonstrate that planar presentations are algorithmically recognizable through combinatorial and graph-theoretic criteria.
• Construct an effective enumeration procedure by systematically generating all valid planar presentations.
• Use the algorithmic recognizability of planar presentations to yield a recursive enumeration of all planar Cayley graphs.

Experimental results

Research questions

• RQ1 Can the class of groups with planar, finitely generated Cayley graphs be effectively enumerated?
• RQ2 What structural properties of group presentations ensure that the associated Cayley graph is planar?
• RQ3 Is there a decidable characterization of planar Cayley graphs in terms of group presentations?
• RQ4 Can the enumeration of planar Cayley graphs be made algorithmic and systematic?

Key findings

• A group admits a planar, finitely generated Cayley graph if and only if it admits a planar presentation.
• Planar presentations are algorithmically recognizable, enabling effective decision procedures for membership in the class of such groups.
• The set of all planar Cayley graphs is effectively enumerable, providing a constructive enumeration procedure.
• The result affirms an open question posed by Droms et al. regarding the effective enumerability of planar groups.