Nonempty intersection of longest paths in graphs without forbidden pairs

In 1966, Gallai asked whether all longest paths in a connected graph have nonempty intersection. The answer to this question is not true general and various counterexamples been found. However, there positive solution Gallai’s for many well-known classes of graphs such as split graphs, series–parallel 2K2-free graphs. Split were proven be Hamiltonian under given toughness conditions. This observation motivates us investigate that are close certain Hamiltonicity properties. Let {R,S} pair particular, paper, we show has an affirmative {R,S}-free every 2-connected Hamiltonian. These pairs completely characterized the 1990s.