A construction of sequentially Cohen-Macaulay graphs

For every simple graph $G$, a class of multiple clique cluster-whiskered graphs $G^{md}$ is introduced, and it is shown that all graphs $G^{md}$ are vertex decomposable, thus the independence simplicial complex ${\rm Ind}\,G^{md}$ is sequentially Cohen-Macaulay; the properties of the graphs $G^{md}$ and the clique-whiskered graph $G^\pi$ are studied, including the enumeration of facets of the complex ${\rm Ind}\, G^{\pi}$ and, the calculation of Betti numbers of the cover ideal $I_c(G^{md})$.