We confirm a conjecture, due to Grötschel, regarding the
intersection vertices of two longest cycles in a graph. In particular,
we show that if G is a graph of circumference at least $k+1$, where
$k\in\{6,7\}$, and G has two longest cycles meeting in a set W of
k vertices, then W is an articulation set. Grötschel had
previously proved this result for $k\in\{3,4,5\}$ and shown that it
fails for $k > 7$. As corollaries, we obtain results regarding the
minimum lengths of longest cycles in certain vertex-transitive graphs.
Our proofs are novel in that they make extensive use of a computer,
although the programs themselves are straightforward