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

Author

Listed:
  • Debajyoti Mondal

    (Bangladesh University of Engineering and Technology (BUET))

  • Rahnuma Islam Nishat

    (Bangladesh University of Engineering and Technology (BUET))

  • Sudip Biswas

    (Bangladesh University of Engineering and Technology (BUET))

  • Md. Saidur Rahman

    (Bangladesh University of Engineering and Technology (BUET))

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.

Suggested Citation

  • Debajyoti Mondal & Rahnuma Islam Nishat & Sudip Biswas & Md. Saidur Rahman, 2013. "Minimum-segment convex drawings of 3-connected cubic plane graphs," Journal of Combinatorial Optimization, Springer, vol. 25(3), pages 460-480, April.
    • Handle: RePEc:spr:jcomop:v:25:y:2013:i:3:d:10.1007_s10878-011-9390-6
    DOI: 10.1007/s10878-011-9390-6
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