The planar Cayley graphs are effectively enumerable II.

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. [ABSTRACT FROM AUTHOR]