Truncation symmetry type graphs

There are operations that transform a map M (an embedding of a graph on a surface) into another map in the same surface, modifying its structure and consequently its set of flags F(M). For instance, by truncating all the vertices of a map M, each flag in F(M) is divided into three flags of the truncated map. Orbanic, Pellicer and Weiss studied the truncation of k-orbit maps for k < 4. 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-, 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 < 8 and k=9

Medial symmetry type graphs

A $k$-orbit map is a map with its automorphism group partitioning the set of flags into $k$ orbits. Recently $k$-orbit maps were studied by Orbani\' c, Pellicer and Weiss, for $k \leq 4$. In this paper we use symmetry type graphs to extend such study and classify all the types of $5$-orbit maps, as well as all self-dual, properly and improperly, symmetry type of $k$-orbit maps with $k\leq 7$. Moreover, we determine, for small values of $k$, all types of $k$-orbits maps that are medial maps. Self-dualities constitute an important tool in this quest

Generation of various classes of trivalent graphs

It turns out that there exist numerous useful classes of cubic graphs. Some are needed in connection with maps, hypermaps, configurations, polytopes, or covering graphs. In this paper, we briefly explore these connections and give motivation why some classes of cubic graphs should be generated. Then we describe the algorithms we used to generate these classes. The results are presented in various tables