Description
We define generalized action graphs as semi-directed graphs in which the edge set is partitioned into directed 2-factors (forming an action digraph) and undirected 1-factors (forming a monodromy graph) and use them to describe several combinatorial structures, such as maps and oriented maps. The quotient of the action graph with respect to its automorphism group (or some of its subgroup) is called the symmetry type graph and is very useful in connection with map symmetries and orientation preserving symmetries. Several usual cases of regular, edge-transitive, vertex-transitive, chiral, etc. maps and oriented maps are revisited. Our symmetry type graphs are closely related to Delaney-Dress symbols and orbifolds. The theory is very general and applies to a variety of discrete structures such as hypermaps, abstract polytopes and maniplexes.
Primary author
Prof.
Tomaž Pisanski
(University of Primorska and University of Ljubljana)
