On a Conjecture for Parameterized st-Orientations

2026-06-12

AI Summary

This study addresses the conjecture by Papamanthou and Tollis regarding st-orientations of biconnected graphs, which posits that the longest path length produced by the MaxSTN algorithm is never shorter than that yielded by the MinSTN algorithm. By integrating graph-theoretic analysis of the MaxSTN and MinSTN algorithms with exhaustive search and constructive proof techniques, the authors discover and rigorously verify a counterexample: a biconnected graph on nine vertices for which MinSTN generates a longest path of length 7, while MaxSTN yields one of length only 6 (i.e., ℓ_max = 6 < ℓ_min = 7). This finding conclusively refutes the universality of the conjecture and provides a critical counterexample that advances theoretical understanding of st-orientations.

Abstract

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.

Problem

• st-orientation
• biconnected graph
• longest path
• conjecture
• graph algorithm

Innovation

• st-orientation
• biconnected graph
• longest path
• conjecture disproof
• graph algorithm