Quotients of Koszul Algebras and 2-D-Determined Algebras
Communications in Algebra
Vatne  and Green and Marcos  have independently studied the Koszul-like homological properties of graded algebras that have defining relations in degree 2 and exactly one other degree. We contrast these two approaches, answer two questions posed by Green and Marcos, and find conditions that imply the corresponding Yoneda algebras are generated in the lowest possible degrees.
Cassidy, Thomas and Phan, Christopher. "Quotients of Koszul Algebras and 2-D-Determined Algebras." Communications in Algebra 42, no. 9 (2014) : 3742-3752.