Optimal st-Orientations for Plane Triangulations

  • Authors:

  • Huaming Zhang;Xin He

  • Affiliations:

  • Computer Science Department, University of Alabama in Huntsville, Huntsville, AL, 35899, USA;Department of Computer Science and Engineering, SUNY at Buffalo, Buffalo, NY, 14260, USA

  • Venue:
  • AAIM '07 Proceedings of the 3rd international conference on Algorithmic Aspects in Information and Management
  • Year:
  • 2007

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Abstract

For a plane triangulation Gwith nvertices, it has been proved that there exists a plane triangulation Gwith nvertices such that for any st-orientation of G, the length of the longest directed paths of Gfrom sto tis $\geq \lfloor \frac{2n}{3}\rfloor$ [18] . In this paper, we prove the bound $\frac{2n}{3}$ is optimal by showing that every plane triangulation Gwith 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 Gwith nvertices admits a visibility representation with height $\le \frac{2n}{3}+O(1)$, constructible in linear time, which is also optimal.