MaxSTN and MinSTN -- proposed by Papamanthou and Tollis (TCS 2008, JGAA 2010) -- are two algorithms for producing $st$-orientations of biconnected graphs with long and short longest paths respectively. Based on extensive experiments on planar and non-planar graphs of up to 5,000 nodes, it was conjectured that $\ell_{\max} \geq \ell_{\min}$ for every biconnected graph $G$, where $\ell_{\max}$ and $\ell_{\min}$ denote the longest-path lengths of the two orientations. This paper disproves this conjecture by exhibiting a biconnected graph on 9 vertices for which MaxSTN yields $\ell_{\max}=6$ while MinSTN yields $\ell_{\min}=7$, regardless of how ties are broken in either algorithm.

翻译：MaxSTN与MinSTN——由Papamanthou和Tollis提出（TCS 2008, JGAA 2010）——是两种用于生成双连通图$st$-定向的算法，分别产生最长路径较长和较短的定向结果。基于对多达5000个节点的平面图与非平面图的大量实验，曾猜想对于每个双连通图$G$，有$\ell_{\max} \geq \ell_{\min}$，其中$\ell_{\max}$与$\ell_{\min}$分别表示两种定向的最长路径长度。本文通过展示一个9顶点双连通图否定了这一猜想：在该图上，无论两个算法中的平局如何处理，MaxSTN给出的$\ell_{\max}=6$，而MinSTN给出的$\ell_{\min}=7$。