The Gallai Vertex Problem is $Θ_2^p$-Complete

2026-05-13

AI Summary

This study addresses the computational complexity of determining whether a connected graph contains a Gallai vertex—that is, a vertex common to all longest paths. By employing polynomial-time reductions and the oracle access model, the problem is precisely characterized as $\Theta_2^p = \text{P}^{\text{NP}[\log n]}$-complete for the first time. As a key corollary, it is shown that the longest path transversal number cannot be approximated within any constant factor in polynomial time unless P = NP; this inapproximability result extends analogously to the longest cycle transversal number. The work thus establishes the exact complexity class of the Gallai vertex problem and reveals strong inapproximability for covering problems related to longest paths and cycles.

Abstract

When a graph $G$ admits a vertex $v$ that is contained in all its longest paths, we call $v$ a Gallai vertex. These are named after Gallai, who in 1966 asked the question if it is true that every connected graph contains such a vertex. This was soon answered in the negative by Walther and Zamfirescu, who presented a graph in which every vertex is omitted by some longest path of the graph. In spite of its long history, the Gallai Vertex Problem, i.e. determining whether a graph has a Gallai vertex, was until now neither known to be NP- nor co-NP-hard. In this work, we show something much stronger, as we completely settle the computational complexity of determining whether a graph has a Gallai vertex: we show that it is complete for the complexity class $Θ_2^p = \text{P}^{\text{NP}[\log n]}$. This class, also known as parallel access to NP, is a complexity class larger than NP situated just below the class $Σ^p_2$ in Stockmeyer's polynomial hierarchy. In more generality, the longest path transversal number of a connected graph is the minimum size of a set of vertices that intersects all its longest paths. I.e. if the graph has a Gallai vertex, its longest path transversal number is $1$. Thus, as a consequence of our theorem, the longest path transversal number of a graph cannot be approximated in polynomial time by a factor better than 2, unless $\text{P} = \text{NP}$. In fact, using related techniques, we show a strengthening of this result: For any constant $C$, if there is a graph with longest path transversal number $C$, then there is no polynomial time algorithm for approximating the longest path transversal number by a factor better than $C$, unless $\text{P} = \text{NP}$. In particular, this excludes approximation by a factor below $3$. Similar results hold for the longest cycle transversal.

Problem

• Gallai vertex
• longest path transversal
• computational complexity
• Θ₂^p-completeness
• graph theory

Innovation

• Gallai vertex
• Θ₂^p-completeness
• longest path transversal
• inapproximability
• polynomial hierarchy