In this paper, we introduce the concept of an intuitionistic fuzzy graph structure (IFGS). We discuss certain notions, including intuitionistic fuzzy $B_i$-cycles, intuitionistic fuzzy $B_i$-trees and $\phi$-complement of an intuitionistic fuzzy graph structure with several examples. We also present $\phi$-complement of an intuitionistic fuzzy graph structure along with self-complementary and strong self-complementary intuitionistic fuzzy graph structures.