A trans-continental railway traverses an entire continent. The railway connecting San Francisco on the West Coast to New York on the East Coast in the United States serves as a prime illustration of a trans-continental railway line.
Option A: Yokohama and Vancouver are situated on different continents and are separated by the Pacific Ocean, rendering a rail connection nonexistent.
Option B: Auckland and Honolulu are located on separate islands, precluding any possibility of a rail connection.
Option C: Correct — San Francisco and New York are linked by one of the world's inaugural and most renowned trans-continental railways.
Option D: London and Aden are not on the same continent and lack any rail connectivity.
Therefore, the correct option is (C): The connection between San Francisco and New York.