Medial Symmetry Type Graphs

  • Isabel Hubard
  • María del Río Francos
  • Alen Orbanić
  • Tomaž Pisanski
Keywords: Symmetry type graph, Medial map, k-orbit map, Flag graph


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 Orbanić, 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.
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