Explanation:

- The titles 'Rayagajakesari' and 'Dayagajakesari' are strongly connected to the Gajapati dynasty.

- The Gajapatis ruled over parts of present-day Odisha and Andhra Pradesh, and their rulers often bore titles with 'Kesari' (meaning lion) and 'Gaja' (meaning elephant), signifying strength and royalty.

- Kakatiya dynasty (Option 1) was another South Indian dynasty, but did not use these specific titles.

- Vijayanagara dynasty (Option 2) had grand titles, but not 'Rayagajakesari' or 'Dayagajakesari'.

- Rashtrakuta dynasty (Option 4) was prominent in the Deccan, but these titles are not found in their lineage.

- The correct option is:

Option 3: Gajapati