A construction of sequentially Cohen-Macaulay graphs

Abstract

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 $\mathrm{Ind}{G}^{md}$ 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 $\mathrm{Ind}{G}^{\pi }$ and, the calculation of Betti numbers of the cover ideal $I_c(G^{md})$.

14 pages; the main construction and the main rsults are improved; the part on strong shellable property is removed fromthis paper