Distance and intersection number in the curve graph of a surface

In this work, we study the cellular decomposition of $S$ induced by a filling pair of curves $v$ and $w$, $Dec_{v,w}(S) = S - (v \cup w)$, and its connection to the distance function $d(v,w)$ in the curve graph of a closed orientable surface $S$ of genus $g$. Efficient geodesics were introduced by the first author in joint work with Margalit and Menasco in 2016, giving an algorithm that begins with a pair of non-separating filling curves that determine vertices $(v,w)$ in the curve graph of a closed orientable surface $S$ and computing from them a finite set of efficient geodesics. We extend the tools of efficient geodesics to study the relationship between distance $d(v,w)$, intersection number $i(v,w)$, and $Dec_{v,w}(S)$. The main result is the development and analysis of particular configurations of rectangles in $Dec_{v,w}(S)$ called spirals. We are able to show that, in some special cases, the efficient geodesic algorithm can be used to build an algorithm that reduces $i(v,w)$ while preserving $d(v,w)$. At the end of the paper, we note a connection of our work to the notion of extending geodesics.