  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.
