Graph Minors .XII. Distance on a Surface

Let Γ a graph drawn on a connected surface Σ which is not a sphere. It is "θ-representative" if every non-null-homotopic closed curve meets Γ at least θ times. Also, Γ defines a metric on Σ, discussed in an earlier paper. Our objective here is to study the effect on the metric and on the "representativeness" of making local changes in the drawing or in the surface. We also reformulate more compactly the main theorem of an earlier paper in terms of this metric. These are lemmas to be used later.