Graphs with Nonempty Intersection of Longest Paths

We prove that the intersection of longest paths in a connected graph \(G\) is nonempty if and only if for every block \(B\) of \(G\) the longest paths in \(G\) which use at least one edge of \(B\) have nonempty intersection. This result is used to show that if every block of a graph \(G\) is Hamilton-connected, almost-Hamilton-connected, or a cycle then all longest paths in \(G\) intersect. (We call a bipartite graph almost-Hamilton-connected if every pair of vertices is connected by a path containing an entire bipartition set.) We also show that in a split graph all longest paths intersect. (A graph is split if there exists a partition of its vertex set into a stable set and a complete set.)