Intersection of Longest Paths in Graph Classes

Let G be a graph and lpt(G) be the size of the smallest set S⊆V(G) such that every longest path of G has at least one vertex in S. If lpt(G) = 1, then all longest paths of G have non-empty intersection. In this work, we prove that this holds for some graph classes, including ptolemaic graphs, P4-sparse graphs, and starlike graphs, generalizing the existing result for split graphs.