A Construction of Sequentially Cohen-Macaulay Graphs

For every simple graph G,a class of multiple clique cluster-whiskered graphs G^(eπm)is introduced,and it is shown that all such graphs are vertex decomposable;thus,the independence simplicial complex Ind G^(eπm)is sequentially Cohen-Macaulay.The properties of the graphs G^(eπm)and G^(π)constructed by Cook and Nagel are studied,including the enumeration of facets of the complex Ind G^(π)and the calculation of Betti numbers of the cover ideal Ic(G^(eπm).We also prove that the complex△=IndH is strongly shellable and pure for either a Boolean graph H=Bn or the full clique-whiskered graph H=G^(W)of C,which is obtained by adding a whisker to each vertex of G.This implies that both the facet ideal I(△)and the cover ideal Ic(H)have linear quotients.