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 ∪ w) , and its connection to the distance function d ( v , w ) in the curve graph of a closed orientable surface S of genus g . Building on the work of Leasure, 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, with appropriate restrictions, 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 between our work and the notion of extending geodesics.