The Doph

The Doph is a Het ship between The Duke and Toph Beifong in the Avatar: The Last Airbender series.

Canon
Toph and The Duke met at some point while Aang was in a coma. The Duke joined Team Avatar on a Fire Nation ship that they had captured. Eventually, another Fire Nation ship attacked. Toph helped fight with her earthbending but could not see due to her blindness so The Duke helped her aim her attacks. The two went separate ways but would later reunite at the Fire Nation invasion. Toph felt sick while she was inside of a submarine and was about to throw up so The Duke took off his helmet and allowed her to vomit in it.

The invasion ended up being a failure so Team Avatar and The Duke escaped on Appa and sought refuge in the Western Air Temple. Toph eventually went missing after leaving to confront Zuko so The Duke suggested that they look for her until she found her way back to the camp. They were separated again when Azula attacked the temple but would reunite at Zuko's coronation after the end of the war. The Duke hugged Toph which surprised her and Pipsqueak held them both on his shoulders.

Fanon
The Doph is a fairly popular ship in the Avatar fandom. It is heavily based on the fact that The Duke seemed to admire Toph and they got along well. Toph was the only member of Team Avatar who was single at the end of the series and many fans had an issue with that. They felt that The Duke was the best possible match for Toph and decided to pair them together. When the Legend of Korra was announced, many fans hoped that The Duke would be Lin's father. The Doph is usually shipped along with the main canon ships which are Kataang, Maiko and Sukka as a solution to Toph being left out. It commonly rivals the Tokka, Taang, Toko, Teoph and Satoph ships.

Trivia

 * In the commentary of the series finale, the creators joked that Toph and The Duke would be cute as a couple.