Operations on Oriented Maps

Abstract

A map on a closed surface is a two-cell embedding of a finite connected graph. Maps on surfaces are conveniently described by certain trivalent graphs, known as flag graphs. Flag graphs themselves may be considered as maps embedded in the same surface as the original graph. The flag graph is the underlying graph of the dual of the barycentric subdivision of the original map. Certain operations on maps can be defined by appropriate operations on flag graphs. Orientable surfaces may be given consistent orientations, and oriented maps can be described by a generating pair consisting of a permutation and an involution on the set of arcs (or darts) defining a partially directed arc graph. In this paper we describe how certain operations on maps can be described directly on oriented maps via arc graphs.Keywords: map; oriented map; truncation; dual; medial; snub; flag graph; arc graph MSC: 52C20, 05C10, 51M20 Maps and Oriented MapsA map M on a closed surface is a two-cell embedding of a finite connected graph G = (V, E) (see, e.g., [1] or [2]). Equivalently, every map can be viewed as a set of three fixed-point-free involutions r 0 , r 1 and r 2 acting on a set of flags Ω with the property that r 0 r 2 is also a fixed-point-free involution, in which case we denote the map as M = (Ω, r 0 , r 1 , r 2 ); see, for instance, [3] (p. 415). With this second point of view, each map can be described completely using a three-edge colored cubic graph, called the flag graph, whose vertices are elements of Ω, where ω 1 and ω 2 are connected by an edge colored i in the flag graph if and only if r i (ω 1 ) = ω 2 . Generally, only connected flag graphs are considered. The graph G is called the skeleton or the underlying graph of a map M.In some cases, one may relax the conditions and allow fixed points in some of the involutions r 0 , r 1 and r 2 . Such a structure describes a map in a surface with a boundary and with its flag graph containing semi-edges. We call such a map and respective flag graph degenerate.If the surface in which the map resides is orientable, then M is said to be an orientable map. There is a well-known test involving flag graphs to determine whether a given map is orientable: Proposition 1. A map is orientable if and only if its flag graph is bipartite.In a bipartite flag graph, the flags (i.e., vertices) of the map come in two color classes; the vertices in a single color class of flags are called arcs. Restricting attention to one set of arcs corresponds to choosing an orientation of the orientable map, and we call the restricted graph an oriented map; each orientable map gives rise to two oppositely oriented oriented maps. We note that in a map there are four flags per edge of the skeleton, while in an oriented map there are two arcs per edge. That is,