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Let p>3 be a prime and let E, E' be supersingular elliptic curves over F_p. We want to construct an isogeny phi: E --> E'. The currently fastest algorithm for finding isogenies between supersingular elliptic curves solves this problem by performing a "meet-in-the-middle" breadth-first search in the full supersingular 2-isogeny graph over F_p^2. In this paper we consider the structure of the isogeny graph of supersingular elliptic curves over F_p. We give an algorithm to construct isogeniesarXiv:1310.7789v1 fatcat:pvbpmuwzrvgoncxbqfzaje72nm