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.
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A preliminary version of this paper was presented at AAIM 2007.
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Zhang, H., He, X. Optimal st-orientations for plane triangulations. J Comb Optim 17, 367–377 (2009). https://doi.org/10.1007/s10878-007-9119-8
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DOI: https://doi.org/10.1007/s10878-007-9119-8