We study the poset G of all unlabelled graphs with H≤G if H occurs as an induced subgraph in G. We present some general results on the Möbius function of intervals of G and some results for specific classes of graphs. This includes a case where the Möbius 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 Möbius function is unbounded. A classification of the disconnected intervals of G is presented, which gives a large class of non-shellable intervals. We also present several conjectures on the structure of G.
Smith, J. P. (2019). The poset of graphs ordered by induced containment. Journal of Combinatorial Theory. Series A, 168, 348–373. https://doi.org/10.1016/j.jcta.2019.06.009