Minimal examples of Cohen-Macaulay independence complexes that fail to be shellable or vertex decomposable

We report on an exhaustive census of all 1,006,700,565 connected graphs on $11$ vertices to find the minimal examples of graphs whose independence complexes are Cohen-Macaulay but not shellable, or shellable but not vertex decomposable. In particular, there are exactly two graphs $G$ where $\mathrm{I}\mathrm{n}\mathrm{d}(G)$ is Cohen-Macaulay but not shellable, when the characteristic is not two, and exactly six graphs where $\mathrm{I}\mathrm{n}\mathrm{d}(G)$ is shellable but not vertex decomposable.