Title: Contractads -- operads with inputs indexed by graphs

Abstract: We introduce a generalization of the notion of an operad that we call a contractad, whose set of operations are indexed by connected graphs and composition rules are numbered by contractions of connected subgraphs, what explains the nature of terminology. We show that many classical operads such as the operad of commutative algebras, Lie algebras, associative algebras, pre-Lie algebras, the little discs operad and the operad of the moduli spaces $\overline{\mathcal{M}}_{0,n}$ of stable curves admits generalizations to contractads. We explain that standard tools like Koszul duality and Gr\"obner bases machinery can be easily generalized to contractads. We verify the koszulness of the commutative, Lie, associative, and Gerstenhaber contractads.