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, P 4 -sparse graphs, and starlike graphs, generalizing the existing result for split graphs. [ABSTRACT FROM AUTHOR]