There are many theorems about connectedness and other such properties of the curve complex of the 5-times punctured sphere. However, I cannot find any explicit descriptions of the curve-complex. (By explicit, I probably mean information that could mindlessly compute a "drawing" of its 1-skeleton in 2 or 3 dimensional space.)
My attempts to do this myself are as follows: Consider the disk in the plane with four punctures placed symmetrically (say $(1,0), (-1,0), (0,1), (0,-1)$). (This is topologically a sphere with five punctures). Using the geometry of the plane, each isotopy class of essential simple closed curves has a "length" defined as the infimum of lengths of curves in the isotopy class. Using this notion of length, subgraphs $C_r$ of the curve complex $C$ can be drawn: $$C_r = \{v \in C: l(v) \leq r\}.$$ Drawing graphs for small $r$, we obtain the following: $$C_7 = \text{petersen graph}$$ $$C_{11} = \begin{cases}\text{connected graph with } 30 \text{ vertices}\\ \text{girth } 5\\ \text{each vertex contained in a cycle of length } 5\\ 10 \text{ vertices degree } 5, \text{ rest degree } 3.\end{cases}$$ I suspect $C_{11}$ embeds in a $(5,5)$-cage. This is interesting since $C_7$ is the unique $(3,5)$-cage. However, I still feel very distant from knowing what the full curve complex of the 5-times punctured sphere is.