GRAPH MINORS .12. DISTANCE ON A SURFACE

Let rbe a graph drawn on a connected surface Sigma which is not a sphe re. It is ''theta-representative'' if every non-null-homotopic closed curve meets Gamma at least theta times. Also, Gamma defines a metric o n Sigma, 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 mo re compactly the main theorem of an earlier paper in terms of this met ric. These are lemmas to be used later; (C) 1995 Academic Press, Inc.