The Gallai Vertex Problem is $Θ_2^p$-Complete
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arXiv
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| Natura: | Preprint |
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2026
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| _version_ | 1866916015298838528 |
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| author | Nikabadi, Amir Rotenberg, Eva Wulf, Lasse |
| author_facet | Nikabadi, Amir Rotenberg, Eva Wulf, Lasse |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2605_13488 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | The Gallai Vertex Problem is $Θ_2^p$-Complete Nikabadi, Amir Rotenberg, Eva Wulf, Lasse Discrete Mathematics Computational Complexity 68Q25, 05C38, 05D15, 05C69, 68R01, 68Q15 F.2.2; G.2.2 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. |
| title | The Gallai Vertex Problem is $Θ_2^p$-Complete |
| topic | Discrete Mathematics Computational Complexity 68Q25, 05C38, 05D15, 05C69, 68R01, 68Q15 F.2.2; G.2.2 |
| url | https://arxiv.org/abs/2605.13488 |