  We study the poset $\mathcal{G}$ of all unlabelled graphs, up to isomorphism,
with $H\le G$ if $H$ occurs as an induced subgraph in $G$. We present some
general results on the M\"obius function of intervals of $\mathcal{G}$ and some
results for specific classes of graphs. This includes a case where the M\"obius
function is given by the Catalan numbers, which we prove using discrete Morse
theory, and another case where it equals the Fibonacci numbers, therefore
showing that the M\"obius function is unbounded. A classification of the
disconnected intervals of $\mathcal{G}$ is presented, which gives a large class
of non-shellable intervals. We also present several conjectures on the
structure of $\mathcal{G}$.
