I am wondering if there are "elementary" ways to check for $n$-connectedness of simplicial complexes in terms of simple conditions on vertices.

For example

Let $X$ be the curve complex on a surface $S$ (the exact type of curve complex does not matter). That is, the vertices of $X$ correspond to curves on $S$. A $p$-simplex of $X$ consists of $p+1$ distinct, disjoint curves.

I am hoping to get conditions like the following examples:

$X$ is connected iff given a curve $c$, and any other curve $\gamma$, there is a sequence of curves $\gamma = \gamma_1, \ldots \gamma_n = c$ such that $\gamma_i$ is disjoint from $\gamma_{i-1}$ and $\gamma_{i+1}$. ie. This is just the saying there is a path from any vertex to the vertex corresponding to $c$.

For $1$-connected, it might look something like this: $X$ is $1$-connected iff for a fixed curve $c$, given an edge in $(a,b)$ in $X$, ie two disjoint curves, there is a sequence of disjoint curves (edges) $(a,b) = (a_1,b_1), \ldots ,(a_n,b_n)$ such that

EITHER ($a_i = a_{i+1}$ and $b_{i+1}$ is disjoint from both $a_{i-1}$ and $b_{i-1}$) OR ($b_i = b_{i+1}$ and $a_{i+1}$ is disjoint from both $b_{i-1}$ and $a_{i-1}$)

$b_n = c$ OR $a_n = c$

For a picture of what the $1$-connected argument is trying to achieve imagine a closed path and slowly filling in with triangles. For each edge $(a,b)$ we get paths $a_1, \ldots, a_n$ and $b_1, \ldots, b_n$ which bound a disk (the disjointness conditions ensure we can fill in with triangles). Actually, this doesn't quite work completely, since edges $(a,b)$ and $(b,c)$ might give rise to different $b$ paths, so one has to ensure that there is a disjointness condition for adjacent vertices also.

So can anyone point me in the right direction if such conditions exist? While I am not sure about the exact form of the answer I want, it would be good if it were something concrete like my above examples to work with. Or could it just be that this is this not really the right question to consider for connectivity arguments?