Minimal graph in which the intersection of two longest paths is not a separator

TL;DR

This paper proves that for small connected graphs with up to 10 vertices, the intersection of two longest paths always separates the graph, establishing minimality of a known counterexample at 11 vertices.

Contribution

It demonstrates that in graphs with up to 10 vertices, the intersection of two longest paths is always a separator, confirming the minimality of the 11-vertex counterexample.

Findings

For graphs with n ≤ 10 vertices, the intersection of two longest paths is always a separator.

The known counterexample with 11 vertices is minimal, as smaller graphs do not exhibit this property.

The result clarifies the structure of longest paths in small connected graphs.

Abstract

We prove that for a connected simple graph $G$ with $n\le 10$ vertices, and two longest paths $C$ and $D$ in $G$, the intersection of vertex sets $V(C)\cap V(D)$ is a separator. This shows that the graph found previously with $n=11$, in which the complement of the intersection of vertex sets $V(C)\cap V(D)$ of two longest paths is connected, is minimal.