Problem 2.B.2 in Hatcher's Algebraic Topology. Also the text uses $\approx$ for an isomorphism.
Show that $\tilde{H}_i(S^n-X)\approx \tilde{H}_{n-i-1}(X)$ when $X$ is homeomorphic to a finite connected graph.
I also have this theorem.
Proposition 2B.1. (a) For an embedding $h: D^k \to S^n,$ $ \tilde{H}_i(S^n-h(D^k))=0 $ for all $i$. (b) For an embedding $h: S^k \to S^n, $with $k<n$, $ \tilde{H}_i(S^n-h(S^k))$ is $\mathbb{Z}$ for $i=n-k-1$ and $0$ otherwise.
Progress: I am given the hint to first do the case that the graph is a tree. I'm thinking about doing this inductively using Mayers-Vietoris.
I have that $\tilde{H}_{n-i-1}(X) \approx \mathbb{Z}^e$ if $i=n-2.$ and $0$ otherwise where $e$ is the edges in $X/T$ where $T$ is the spanning tree of $X$.