Equivalence of Turn-Regularity and Complete Extensions

The aim of the two-dimensional compaction problem is to minimize the total edge length or the area of an orthogonal grid drawing. The coordinates of the vertices and the length of the edges can be altered while all angles and the shape of the drawing have to be preserved. The problem has been shown to be NP-hard. Two commonly used compaction methods are the turn-regularity approach by (Bridgeman et al., 2000) and the approach by (Klau and Mutzel, 1999) considering complete extensions. We formally prove that these approaches are equivalent, i. e. a face of an orthogonal representation is turn-regular if and only if there exists a unique complete extension for the segments bounding this face.