You'd better sit down for this one. That's right.
Facebook thinks Dokdo belongs to Japan. Well, sort of.
Do a Facebook Places search for "Dokdo" or even "독도" on Facebook, and it returns results saying that this location, while using the neutral/Eurocentric name "Liancourt Rocks," is located in Japan.
There are plenty of Pages and Groups devoted to Dokdo-related propagation (독도는 우리 땅) that come up in the search results, but it looks like the actual Facebook Places entry lists it as being part of Japan. A search for Takeshima brings up this same page. The page clearly lists the islands as being part of Oki-gun, Shimane Prefecture, Japan.
It's a bit weird that this is the first result. When you post a status update, if you add a location ("check in") and type "Dokdo," this is the location that appears first in the list, with just 12 check-ins right now. But the second entry is much more popular, at 1,511 check-ins. And it seems like it's grown since just two days ago.
As for the second Dokdo location in that list, Facebook says it's part of Ulleungdo in Korea, but bizarrely includes in the title 隠岐の島町竹島官有無番地, suggesting that it's Takeshima, again under the Oki Islands, Shimane Prefecture, Japan.
Over on Google Maps, the listing for Dokdo cites it as part of Ulleung-gun, Korea, though labels them Liancourt Rocks.
I couldn't find any Google search result for Takeshima, 竹島, or たけしま that reliably leads to these islands. Of course, I'm using the US-based google.com domain. Switching over to the Japanese domain (google.co.jp), the islands appear as 竹島 (Takeshima).
And on the Korean domain (google.co.kr), they become 독도 (Dokdo):
Maybe Google just gives the audiences what they want. Of course Facebook's Places listings are more easily user-edited or user-contributed than Google's, so Netizen activism might play a role.
If you're interested in more Dokdo mapping, here's a simple but interesting map measuring distances between Dokdo and Korea/Japan.