Double-orientations on supersingular isogeny graphs

We recall and define various kinds of supersingular $\mathrm{\ell }$-isogeny graphs and precise graph isomorphism with a corresponding quaternion $\mathrm{\ell }$-ideal graph. In particular, we introduce the notion of double-orientations on supersingular elliptic curves and study the structure of double-oriented supersingular $\mathrm{\ell }$-isogeny graphs.