Explanation:

Hoysala Dynasty: The Hoysala dynasty ruled most of what is now Karnataka, India, between the 10th and 14th centuries. Their rule began in the 10th century and reached its peak under kings like Vishnuvardhana and Veera Ballala II.

Gahadavala (or Gaharwar) Dynasty: This dynasty ruled the region of Kannauj (in present-day Uttar Pradesh) during the late 11th and the 12th centuries. Their rule started in the late 11th century.

Kakatiya Dynasty: The Kakatiya dynasty ruled parts of present-day Telangana and Andhra Pradesh in South India from the 12th to the 14th century. Their rule began in the 12th century.

Yadava Dynasty: The Yadava dynasty, also known as Seuna dynasty, ruled a kingdom stretching from the Tungabhadra river in the south to the Narmada river in the north in present-day Maharashtra and parts of Madhya Pradesh. Their ascendancy to power is marked from the late 12th century.

From the given options, none of the dynasties established their kingdoms in the early 8th century AD. The correct answer is:

4. None.