The VK Yuri Gagarin was a Terran DY-732-class starship that was in service in the early 22nd century. This ship was commanded by Captain Winrich Kolbe.

In 2105, the Yuri Gagarin was launched from Earth on a colonization mission.

This vessel was among the list of ships that appeared on Captain Jean-Luc Picard's desktop monitor while he searched for "all Earth deep-space launches from 2123 until 2190 with destination in or near the Ficus sector" during an investigation of an SOS signal discovered in that sector in 2365. This vessel, however, did not fall within the parameters of his search, but was among the ships launched in that timeframe. (TNG: "Up The Long Ladder")

Named for Yuri Gagarin, first man in space.
In the book The Hunt for Red October, the primary Alfa-class attack sub was named V.K. Konovalov, in honor of Vladimir Konstantinovich Konovalov. This suggests that the use of "VK" was an homage to Tom Clancy. In the real world, Russians do not use prefixes on their ships, but an in-universe explanation could be that "VK" stands for Voenny Korabl, the romanization of Военный Корабль, Military Ship.
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