Resumen en español

In this paper we study the grafted simplicial complex, introduced by Sara Faridi, as higher-dimensional graphs. We obtain combinatorial properties of interest of this class of complexes, which allow us to demonstrate that the property of a simplicial complex being Cohen-Macaulay is related to certain subcomplexes of it. As a consequence we obtain an alternative proof of Faridi's theorem: every grafted facet complex is Cohen-Macaulay.