Intersection of Longest Paths in a Graph

In 1966, Gallai asked whether every connected graph has a vertex that is common to all its longest paths. The answer to this question is negative. We prove that the answer is positive for outerplanar graphs. Another related question was raised in 1995 at the British Combinatorial Conference: Do any three longest paths in a connected graph have a vertex in common? We prove that, in a connected graph in which all non-trivial blocks are Hamiltonian, any three of its longest paths have a common vertex. Both of these results strengthen a recent result by Axenovich.