$d(u, a) \leq 4k$. So $d(t, u') \leq 4k$ or $d(t, c) \leq 4k$, making the triangle $xyz$ $4k$-thin.

Bowditch shows, in [@Bow] Proposition 3.1, that we don’t always have to work with geodesic triangles to show hyperbolicity of a graph.

\[subset hyperbolic\] Given $h \geq 0$, there exists $\delta \geq 0$ with the following property. Suppose that $G$ is a connected graph, and that for each $x, y \in V(G)$, we have associated a connected subgraph, ${\mathcal{L}}(x,y) \subset G$, with $x, y \in {\mathcal{L}}(x,y)$. Suppose that:

1.  for all $x, y, z \in V(G)$, $${\mathcal{L}}(x,y) \subset N_h({\mathcal{L}}(x,z) \cup {\mathcal{L}}(z, y))$$ and

2.  for any $x, y \in V(G)$ with $d(x,y) \leq 1$, the diameter of ${\mathcal{L}}(x,y)$ in $G$ is at most $h$.

Then $G$ is $\delta$-thin hyperbolic. In fact, we can take any $\delta \geq (3m-10h)/2$, where $m$ is any positive real number satisfying $$2h(6 + \log_2(m+2)) \leq m.$$

Graphs
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Let $S = S_{g,p}$ be a surface where $g$ is the genus and $p$ is the number of punctures. We define $\xi(S_{g,p}) = 3g + p -3$ and refer to $\xi(S_{g,p})$ as the complexity of $S_{g,p}$. When $\xi(S) > 1$ the curve graph of $S$, ${\mathcal{C}}(S)$, originally introduced 