The longest path transversal number of a connected graph $G$ denoted by lpt($G$), is the minimum size of a set of vertices in $G$ that intersects all longest paths of $G$. Surprisingly, it remains widely open whether there exists a constant $c$ such that every connected graph $G$ satisfies $\mathrm{lpt}(G)\leq c$.

We present constant upper bounds on the longest path transversal number for hereditary classes of graphs, that is, classes of graphs closed under taking induced subgraphs. Based on joint work with Paloma T. Lima and Paweł Rzążewski.