The poset of graphs ordered by induced containment

TL;DR

This paper explores the poset of all unlabelled graphs ordered by induced subgraph containment, analyzing the M"obius function and structural properties, revealing connections to Catalan and Fibonacci numbers and discussing conjectures on its structure.

Contribution

It provides new results on the M"obius function of the graph poset, including cases linked to Catalan and Fibonacci numbers, and classifies disconnected intervals, advancing understanding of its combinatorial structure.

Findings

M"obius function equals Catalan numbers in certain cases

M"obius function can be unbounded, related to Fibonacci numbers

Classification of disconnected intervals and non-shellable intervals

Abstract

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 $\mathcal{G}$.