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Optimal st-orientations for plane triangulations

Author

Listed:
  • Huaming Zhang

    (University of Alabama in Huntsville)

  • Xin He

    (SUNY at Buffalo)

Abstract

For plane triangulations, it has been proved that there exists a plane triangulation G with n vertices such that for any st-orientation of G, the length of the longest directed paths of G in the st-orientation is $\geq\lfloor\frac{2n}{3}\rfloor$ (Zhang and He in Lecture Notes in Computer Science, vol. 3383, pp. 425–430, 2005). In this paper, we prove the bound $\frac{2n}{3}$ is optimal by showing that every plane triangulation G with n-vertices admits an st-orientation with the length of its longest directed paths bounded by $\frac{2n}{3}+O(1)$ . In addition, this st-orientation is constructible in linear time. A by-product of this result is that every plane graph G with n vertices admits a visibility representation with height $\le\frac{2n}{3}+O(1)$ , constructible in linear time, which is also optimal.

Suggested Citation

  • Huaming Zhang & Xin He, 2009. "Optimal st-orientations for plane triangulations," Journal of Combinatorial Optimization, Springer, vol. 17(4), pages 367-377, May.
    • Handle: RePEc:spr:jcomop:v:17:y:2009:i:4:d:10.1007_s10878-007-9119-8
    DOI: 10.1007/s10878-007-9119-8
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    Keywords

    st-orientation; Plane triangulation;

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