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 curves v and w, \(Dec_{v,w}(S) = {\setminus } (v \cup w)\), its connection to distance function d(v, w) in curve graph closed orientable surface genus g. Building on work Leasure, efficient geodesics were introduced first author joint with Margalit Menasco 2016, giving an algorithm that begins non-separating determine vertices (v, computing from them finite set geodesics. We extend tools relationship between w), intersection number i(v, \(Dec_{v,w}(S)\). The main result is development analysis particular configurations rectangles \(Dec_{v,w}(S)\) called spirals. are able show that, appropriate restrictions, geodesic can be used build reduces while preserving w). At end paper, note our notion extending