We show that efficient geodesics have the strong property of "super
efficiency". For any two vertices, v,wβC(Sgβ), in the complex
of curves of a closed oriented surface of genus gβ₯2, and any efficient
geodesic, v=v1β,β―,vdβ=w, it was previously established
by Birman, Margalit and the second author (see arXiv:1408.4133) that there is
an explicitly computable list of at most d(6gβ6) candidates for
the v1β vertex. In this note we establish a bound for this computable list
that is independent of d-distance and only dependent on genus---the
super efficiency property. The proof relies on a new intersection growth
inequality between intersection number of curves and their distance in the
complex of curves, together with a thorough analysis of the dot graph
associated with the intersection sequence.Comment: 24 pages, 20 figure