The correct answer is option 'A', Vijayanagara Kings. Let's delve into the details to understand why.

The Vijayanagara Empire was a powerful South Indian empire that emerged during the 14th century CE. It was founded by two brothers, Harihara I and Bukka I, who served as military commanders in the Kakatiya Kingdom. After the decline of the Kakatiya Kingdom, they established their own empire with its capital at Vijayanagara (modern-day Hampi).

During their reign, the Vijayanagara Kings competed with contemporary rulers for control of fertile river valleys. This was primarily due to the strategic importance of river valleys for agriculture, trade, and overall economic prosperity. The fertile river valleys provided ample opportunities for cultivation and irrigation, making them highly sought after by various rulers.

The Vijayanagara Kings engaged in territorial expansion and military campaigns to assert their control over these fertile regions. They faced fierce competition from neighboring kingdoms, such as the Bahmani Sultanate in the Deccan, the Gajapati Kingdom in Odisha, and the various Nayaka rulers in Tamil Nadu and Karnataka.

The Vijayanagara Kings implemented effective administrative and military strategies to consolidate their power. They established a decentralized system of governance, dividing the empire into administrative units known as Nadu, with each Nadu being governed by a local chief. This allowed for efficient administration and ensured the smooth functioning of the empire.

Additionally, the Vijayanagara Kings maintained a powerful military force consisting of infantry, cavalry, and elephants. They also had a navy to safeguard their coastal territories. The military prowess of the Vijayanagara Empire played a crucial role in their competition with other rulers for control of fertile river valleys.

Furthermore, the Vijayanagara Kings patronized art, literature, and architecture, resulting in the development of a unique architectural style known as the Vijayanagara style. Numerous temples and monuments were constructed during their reign, showcasing their grandeur and cultural influence.

In conclusion, the Vijayanagara Kings competed with contemporary rulers for control of fertile river valleys. Through their military campaigns, administrative strategies, and patronage of art and architecture, they established themselves as a formidable empire in South India.