Minimum-segment convex drawings of 3-connected cubic plane graphs

A convex drawing of a plane graph G is a plane drawing of G, where each vertex is drawn as a point, each edge is drawn as a straight line segment and each face is drawn as a convex polygon. A maximal segment is a drawing of a maximal set of edges that form a straight line segment. A minimum-segment convex drawing of G is a convex drawing of G where the number of maximal segments is the minimum among all possible convex drawings of G. In this paper, we present a linear-time algorithm to obtain a minimum-segment convex drawing Γ of a 3-connected cubic plane graph G of n vertices, where the drawing is not a grid drawing. We also give a linear-time algorithm to obtain a convex grid drawing of G on an $(frac{n}{2}+1)times(frac {n}{2}+1)$ grid with at most s _n +1 maximal segments, where $s_{n}=frac{n}{2}+3$ is the lower bound on the number of maximal segments in a convex drawing of G.