The \textit{longest path transversal number} of a connected graph $G$, denoted by $lpt(G)$, is the minimum size of a set of vertices of $G$ that intersects all longest paths in $G$. We present constant upper bounds for the longest path transversal number of \textit{hereditary classes of graphs}, that is, classes of graphs closed under taking induced subgraphs. Our first main result is a structural theorem that allows us to \textit{refine} a given longest path transversal in a graph using domination properties. This has several consequences: First, it implies that for every $t \in \{5,6\}$, every connected $P_t$-free graph $G$ satisfies $lpt(G) \leq t-2$. Second, it shows that every $(\textit{bull}, \textit{chair})$-free graph $G$ satisfies $lpt(G) \leq 5$. Third, it implies that for every $t \in \mathbb{N}$, every connected chordal graph $G$ with no induced subgraph isomorphic to $K_t \mat \overline{K_t}$ satisfies $lpt(G) \leq t-1$, where $K_t \mat \overline{K_t}$ is the graph obtained from a $t$-clique and an independent set of size $t$ by adding a perfect matching between them. Our second main result provides an upper bound for the longest path transversal number in \textit{$H$-intersection graphs}. For a given graph $H$, a graph $G$ is called an \textit{$H$-graph} if there exists a subdivision $H'$ of $H$ such that $G$ is the intersection graph of a family of vertex subsets of $H'$ that each induce connected subgraphs. The concept of $H$-graphs, introduced by Bir\'o, Hujter, and Tuza, naturally captures interval graphs, circular-arc graphs, and chordal graphs, among others. Our result shows that for every connected graph $H$ with at least two vertices, there exists an integer $k = k(H)$ such that every connected $H$-graph $G$ satisfies $lpt(G) \leq k$.

翻译：连通图$G$的\textit{最长路径横截数}，记作$lpt(G)$，是指与$G$中所有最长路径均相交的顶点集的最小大小。本文针对\textit{遗传图类}（即在取诱导子图操作下封闭的图类）的最长路径横截数提出了常数上界。我们的第一个主要结果是一个结构定理，该定理允许我们利用支配性质对给定图中的最长路径横截集进行\textit{细化}。这带来了若干推论：首先，对于任意$t \in \{5,6\}$，每个连通的无$P_t$图$G$满足$lpt(G) \leq t-2$。其次，它表明每个$(\textit{bull}, \textit{chair})$无图$G$满足$lpt(G) \leq 5$。第三，对于任意$t \in \mathbb{N}$，每个不包含与$K_t \mat \overline{K_t}$同构的诱导子图的连通弦图$G$满足$lpt(G) \leq t-1$，其中$K_t \mat \overline{K_t}$是通过在一个$t$团和一个大小为$t$的独立集之间添加完美匹配所得的图。我们的第二个主要结果为\textit{$H$相交图}中的最长路径横截数提供了一个上界。对于给定图$H$，若存在$H$的一个细分$H'$，使得图$G$是$H'$上若干顶点子集的相交图，且每个子集均诱导出连通子图，则称$G$为\textit{$H$图}。由Bir\'o、Hujter和Tuza引入的$H$图概念自然地涵盖了区间图、圆弧图和弦图等图类。我们的结果表明：对于任意至少包含两个顶点的连通图$H$，存在一个整数$k = k(H)$，使得每个连通的$H$图$G$满足$lpt(G) \leq k$。