  In 1966, T. Gallai asked whether every connected graph has a vertex that
appears in all longest paths. Since then this question has attracted much
attention and many work has been done in this topic. One important open
question in this area is to ask whether any three longest paths contains a
common vertex in a connected graph. It was conjectured that the answer to this
question is positive. In this paper, we propose a new approach in view of
distances among longest paths in a connected graph, and give a substantial
progress towards the conjecture along the idea.
