Upward drawing on the plane grid using less ink

Abstract

Any upward drawing D(P) on a two-dimensional integer grid I, of an ordered set P, has completion P with an upward drawing D(P) on a two-dimensional integer grid I such that the total edge length of D(P) does not exceed the total edge length of D(P). Moreover, by (possibly) translating vertices, there is an upward drawing D(P) on I such that I = I. Thus, any integer grid embedding of a two-dimensional ordered set can be extended to a planar upward drawing of its completion, on the same integer grid, without increasing the total edge length.