Curve complex

Remarks

A maximal set of pairwise disjoint, pairwise non-isotopic essential curves has $3g - 3 + n$ elements, a pants decomposition, so $\mathcal{C}(S)$ has dimension $3g - 4 + n$. The mapping class group acts on $\mathcal{C}(S)$, and under the hypothesis $3g - 3 + n \ge 2$ the complex is connected and has infinite diameter.

Example: for the sphere with $5$ boundary circles, $g = 0$ and $n = 5$, so $3g - 3 + n = 2$ and $\mathcal{C}(S)$ is a graph. Each essential curve cuts off two of the boundary circles from the other three, and two such curves are joined by an edge when they can be made disjoint.

Non-example of clause 2: on a closed surface of genus $2$, let curves $a$ and $b$ meet transversally in one point. Their algebraic intersection number is $\pm 1$, so they have no disjoint representatives. So $a$ and $b$ are not joined by an edge.

For the torus, the once-punctured torus and the four-holed sphere, the definition gives a complex without edges. For them one joins curves of minimal intersection by edges instead, and the result is the Farey graph.