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.

翻译：当图 $G$ 中存在一个顶点 $v$ 包含于其所有最长路径中时，我们称 $v$ 为加莱顶点。这一命名源于加莱（Gallai）在1966年提出的问题：是否每个连通图都包含这样的顶点？随后，瓦尔特尔（Walther）和扎姆菲雷斯库（Zamfirescu）很快给出了否定回答，他们构造了一个图，其中每个顶点都被某条最长路径所排除。尽管该问题历史悠久，但加莱顶点问题（即确定一个图是否含有加莱顶点）此前既未被证明属于NP类，也未被证明属于co-NP类。本工作中，我们展示了更强的结论，完全解决了判定图是否存在加莱顶点的计算复杂度问题：我们证明该问题对于复杂度类 $Θ_2^p = \text{P}^{\text{NP}[\log n]}$ 是完备的。该类也称为并行访问NP类，是一个大于NP的复杂度类，在斯托克迈耶（Stockmeyer）的多项式谱系中略低于 $Σ^p_2$ 类。更一般地，连通图的最长路径横贯数定义为与所有最长路径相交的顶点集的最小基数，即若图存在加莱顶点，其最长路径横贯数为 $1$。因此，作为我们定理的推论，除非 $\text{P} = \text{NP}$，否则图的最长路径横贯数无法在多项式时间内以优于2的因子近似。事实上，利用相关技术，我们证明该结果的加强形式：对于任意常数 $C$，若存在一个最长路径横贯数为 $C$ 的图，则除非 $\text{P} = \text{NP}$，否则不存在多项式时间算法能以优于 $C$ 的因子近似最长路径横贯数，特别地，这排除了因子低于3的近似可能性。类似结论对于最长圈横贯数同样成立。