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For the complex of curves of a closed orientable surface of genus $g$,$\mathcal{C}(S_{g>1})$, the notion of efficient geodesic in was introduced inarXiv:1408.4133. There it was established that there always exists (finitelymany) efficient geodesics between any two vertices, $ v_{\alpha} , v_{\beta}\in \mathcal{C}(S_g)$, representing homotopy classes of simple closed curves,$\alpha , \beta \subset S_g$. The main tool for used in establishing theexistence of efficient geodesic was a dot graph, a booking scheme for recordingthe intersection pattern of a reference arc, $\gamma \subset S_g$, with thesimple closed curves associated with the vertices of geodesic path in the zeroskeleton, $\mathcal{C}^0(S_g)$. In particular, for an efficient geodesicbetween $v_\alpha $ and $v_\beta$ of length $d \geq 3$, it was shown that anycurve corresponding to the vertex that is distance one from $v_\alpha$intersects any $\gamma$ at most $d -2$ times. In this note we make a moreexpansive study of the characterizing "shape" of the dot graphs over the entireset of vertices in an efficient geodesic edge-path. The key take away of thisstudy is that the shape of a dot graph for any efficient geodesic is containedwithin a spindle shape region. Since the Nielson-Thurston coordinates of anycurve on $S_g$ are directly derived from its intersection number with finitelymany reference arcs, spindle shaped dot graphs control the coordinate behaviorof curves associated with the vertices of an efficient geodesic.

Efficient geodesics in the curve complex and their dot graphs