The phenomenon which is influenced by auxin and also played a major role in its discovery 

Phototropism is the phenomenon where plants grow or bend towards a light source. It is one of the major responses influenced by the plant hormone auxin. The discovery of auxin was closely associated with studying and understanding phototropism.

The role of auxin in phototropism was first investigated by Charles Darwin and his son Francis Darwin. They conducted experiments with oat seedlings and observed that the tips of the Coleoptiles (the protective sheath covering the emerging shoot) were responsible for bending towards a light source. They hypothesised that a signal was being transmitted from the tip to the elongating region of the coleoptile, causing it to grow towards the light. 

(2) Auxin is a plant growth regulator that plays a significant role in root initiation but it does not play a role in the discovery of auxin. 

(3) Gravitropism, also known as geotropism, is the phenomenon in which plants respond to gravity by growing in a
specific direction. It is another important plant response influenced by auxin. Auxin plays a crucial role in gravitropism
by regulating the differential growth and curvature of plant organs in response to gravity. Gravitropism does not play a
role in the discovery of auxin. 

(4) Apical dominance is a phenomenon in plant growth and development where the main growing tip, or the apical
meristem, of a plant inhibits the growth of lateral buds or branches below it. It is primarily regulated by the plant hormone auxin. Apical dominance did not play a role in the discovery of auxin.