@article{Moradi_Khosh-Ahang_2016, title={On Vertex Decomposable Simplicial Complexes and Their Alexander Duals}, volume={118}, url={https://www.mscand.dk/article/view/23295}, DOI={10.7146/math.scand.a-23295}, abstractNote={In this paper we study the Alexander dual of a vertex decomposable simplicial complex. We define the concept of a vertex splittable ideal and show that a simplicial complex $\Delta$ is vertex decomposable if and only if $I_{\Delta^{\vee }$ is a vertex splittable ideal. Moreover, the properties of vertex splittable ideals are studied. As the main result, it is proved that any vertex splittable ideal has a Betti splitting and the graded Betti numbers of such ideals are explained with a recursive formula. As a corollary, recursive formulas for the regularity and projective dimension of $R/I_{\Delta}$, when $\Delta$ is a vertex decomposable simplicial complex, are given. Moreover, for a vertex decomposable graph $G$, a recursive formula for the graded Betti numbers of its vertex cover ideal is presented. In special cases, this formula is explained, when $G$ is chordal or a sequentially Cohen-Macaulay bipartite graph. Finally, among the other things, it is shown that an edge ideal of a graph is vertex splittable if and only if it has linear resolution. }, number={1}, journal={MATHEMATICA SCANDINAVICA}, author={Moradi, Somayeh and Khosh-Ahang, Fahimeh}, year={2016}, month={Mar.}, pages={43–56} }