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Cohen-Macaulay graphs and face vectors of flag complexes

We introduce a construction on a flag complex that, by means of modifying the associated graph, generates a new flag complex whose $h$-factor is the face vector of the original complex. This construction yields a vertex-decomposable, hence Cohen-Macaulay, complex. From this we get a (non-numerical) characterisation of the face vectors of flag complexes and deduce also that the face vector of a flag complex is the $h$-vector of some vertex-decomposable flag complex. We conjecture that the converse of the latter is true and prove this, by means of an explicit construction, for $h$-vectors of Cohen-Macaulay flag complexes arising from bipartite graphs. We also give several new characterisations of bipartite graphs with Cohen-Macaulay or Buchsbaum independence complexes.