Truncation Symmetry Type Graphs

Abstract

There are operations that transform a map \(\mathcal{M}\) (an embedding of a graph on a surface) into another map on the same surface, modifying its structure and consequently its set of flags \(\mathcal{F(M)}\). For instance, by truncating all the vertices of a map \(\mathcal{M}\), each flag in \(\mathcal{F(M)}\) is divided into three flags of the truncated map. Orbanić, Pellicer, and Weiss studied the truncation of \(k\)-orbit maps for \(k \leq 3\). They introduced the notion of \(T\)-compatible maps in order to give a necessary condition for a truncation of a \(k\)-orbit map to be either \(k\)-, \(\frac{3k}{2}\)-, or \(3k\)-orbit map. Using a similar notion, by introducing an appropriate partition on the set of flags of the maps, we extend the results on truncation of \(k\)-orbit maps for \(k \leq 7\) and \(k = 9\).