A Note on Minimum-Segment Drawings of Planar Graphs

A straight-line drawing of a planar graph G is a planar drawing of G, where each vertex is mapped to a point on the Euclidean plane and each edge is drawn as a straight line segment. A segment in a straight-line drawing is a maximal set of edges that form a straight line segment. A minimum-segment drawing of G is a straightline drawing of G, where the number of segments is the minimum among all possible straight-line drawings of G. In this paper we prove that it is NP-complete to determine whether a plane graph G has a straight-line drawing with at most k segments, where k ≥ 3. We also prove that the problem of deciding whether a given partial drawing of G can be extended to a straight-line drawing with at most k segments is NP-complete, even when G is an outerplanar graph. Finally, we investigate a worst-case lower bound on the number of segments required by straight-line drawings of arbitrary spanning trees of a given planar graph.