| Abstract: | Let G be a connected graph with edge set E embedded in the surface ∑. Let G° denote the geometric dual of G. For a subset d of E, let τd denote the edges of G° that are dual to those edges of G in d. We prove the following generalizations of well-known facts about graphs embedded in the plane. (1) b is a boundary cycle in G if and only if τb is a cocycle in G°. (2) If T is a spanning tree of G, then τ(E/T) contains a spanning tree of G°. (3) Let T be any spanning tree of G and, for e ? E/T, let T(e) denote the fundamental cycle of e. Let U ∪ E/T. Then τU is a spanning tree of G° if and only if the set of face boundaries, less any one, together with the set {T(e); e ? E/(T ∪ U)} is a basis for the cycle space of G. |
