Story of Demoness Ayomukhi and Lakshman is found in the Aranya Kanda of Ramayana. Sita is kidnapped and the two brothers start a frantic search for her. They meet Jatayu who was mortally injured by Ravana. He gives them the direction in which Ravana carried Sita away.
The brothers were now walking towards South. The passed through the dense
they reached a forest named Krauncharanya. Here they came across a deep cave.
In the mouth of the cave they saw a Rakshasi named Ayomukhi. forest
The demoness was huge in size and fearful to look at. She approached Lakshman and expressed her desire to marry him.
Lakshman who was tormented by the kidnapping of Sita had no patience. He took out his sword and severed her nose, ears and breasts.
The Rakshasi ran away from the spot.