= Solution
The <curve complex> $\mathcal C(S)$ has one vertex for each isotopy class of essential simple closed curves, and a set of vertices spans a simplex when it has pairwise disjoint representatives.
To prove connectedness, put two curves $\alpha,\beta$ in minimal position and induct on $i(\alpha,\beta)$. If it is zero, their vertices are equal or joined by an edge. If it is positive, surgery of $\alpha$ along an outermost segment of $\beta$ produces a boundary component $\gamma$ of a regular neighborhood of the surgery. At least one choice is essential; it is disjoint from $\beta$ and satisfies
$$
i(\alpha,\gamma)<i(\alpha,\beta).
$$
The induction hypothesis gives a path from $\alpha$ to $\gamma$, and the edge from $\gamma$ to $\beta$ completes it. Thus $\mathcal C(S)$ is connected. This is the <connectedness of the curve complex>.
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