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

Authors:
Sudip Biswas;Debajyoti Mondal;Rahnuma Islam Nishat;Md. Saidur Rahman
Affiliations:
Department of Computer Science and Engineering, Bangladesh University of Engineering and Technology, Bangladesh;Department of Computer Science and Engineering, Bangladesh University of Engineering and Technology, Bangladesh;Institute of Information and Communication Technology, Bangladesh University of Engineering and Technology, Bangladesh;Department of Computer Science and Engineering, Bangladesh University of Engineering and Technology, Bangladesh
Venue:
COCOON'10 Proceedings of the 16th annual international conference on Computing and combinatorics
Year:
2010

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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 lineartime 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 (n/2 + 1) × (n/2 +1) grid with at most sn + 1 maximal segments, where sn = n/2 + 3 is the lower bound on the number of maximal segments in a convex drawing of G.