Nonempty intersection of longest paths in $2K_2$-free graphs

TL;DR

This paper proves that in nonempty $2K_2$-free graphs, all longest paths share at least one common vertex, extending known results for specific graph classes and providing a new proof for split graphs.

Contribution

It establishes that every vertex of maximum degree in nonempty $2K_2$-free graphs lies on all longest paths, ensuring their intersection.

Findings

All longest paths in nonempty $2K_2$-free graphs intersect.

Vertices of maximum degree are on all longest paths.

Provides a new proof for the intersection property in split graphs.

Abstract

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.