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 $\mathcal{F}\mathcal{(}\mathcal{M}\mathcal{)}$. For instance, by truncating all the vertices of a map $\mathcal{M}$, each flag in $\mathcal{F}\mathcal{(}\mathcal{M}\mathcal{)}$ is divided into three flags of the truncated map. Orbanić, Pellicer, and Weiss studied the truncation of $k$-orbit maps for $k\le 3$. 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 $k\le 7$ and $k=9$.