The poset of graphs ordered by induced containment

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 $\mathcal{G}$.