Nonempty intersection of longest paths in 2K₂-free graphs

In 1966, Gallai asked whether all longest paths in a connected graph share a common vertex. Counterexamples indicate that this is not true in general. However, Gallai's question is positive for certain well-known classes of connected graphs, such as split graphs, interval graphs, circular arc graphs, outerplanar graphs, and series- parallel graphs. A graph is 2K₂-free if it does not contain two independent edges as an induced subgraph. In this short note, we show that, in nonempty 2K₂-free graphs, every vertex of maximum degree is common to all longest paths. Our result implies that all longest paths in a nonempty 2K₂-free graph have a nonempty intersection. In particular, it strengthens the result on split graphs, as split graphs are 2K₂-free.