  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 $2K_2$-free if it
does not contain two independent edges as an induced subgraph. In this paper,
we show that in nonempty $2K_2$-free graphs, every vertex of maximum degree is
common to all longest paths. Our result implies that all longest paths in a
nonempty $2K_2$-free graph have a nonempty intersection. In particular, it
gives a new proof for the result on split graphs, as split graphs are
$2K_2$-free.
