This paper is about a type of quantitative density of closed geodesics and orthogeodesics on complete finite-area hyperbolic surfaces. The main results are upper bounds on the length of the shortest closed geodesic and the shortest doubly truncated orthogeodesic that are $\varepsilon$-dense on a given compact set on the surface.

It is a theorem of Bers that any closed hyperbolic surface admits a pants decomposition consisting of curves of bounded length where the bound only depends on the topology of the surface. The question of the quantification of the optimal constants has been well studied, and the best upper bounds to date are linear in genus, due to a theorem of Buser and Seppälä. The goal of this note is to give a short proof of a linear upper bound that slightly improves the best known bound.

Let M be a hyperbolic surface and Γ(M) its extended mapping class group. We show that Γ(M) is isomorphic to the automorphism group of the following graph G(M). The set of vertices of G(M) is the set S(M) of nonseparating simple closed geodesics of M. Two vertices u and v of S(M) are related by an edge if u and v intersect exactly once in M. The graph G(M) can be thought of as a combinatorial model for M.