Abstract
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.
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Mondal, D., Nishat, R.I., Biswas, S. et al. Minimum-segment convex drawings of 3-connected cubic plane graphs. J Comb Optim 25, 460–480 (2013). https://doi.org/10.1007/s10878-011-9390-6
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DOI: https://doi.org/10.1007/s10878-011-9390-6