  Let $K$ be a field and $R=K[x_1,\ldots, x_n]$ be the polynomial ring in $n$
variables over a field $K$. Let $\Delta$ be a simplicial complex on $n$
vertices and $I=I_{\Delta}$ be its Stanley-Reisner ideal. In this paper, we
show that if $I$ is a matroidal ideal then the following conditions are
equivalent: $(i)$ $\Delta$ is sequentially Cohen-Macaulay; $(ii)$ $\Delta$ is
shellable; $(iii)$ $\Delta$ is vertex decomposable. Also, if $I$ is a minimally
generated by $u_1,\ldots,u_s$ such that $s\leq 3$ or
$\supp(u_i)\cup\supp(u_j)=\{x_1,\ldots,x_n\}$ for all $i\neq j$, then $\Delta$
is vertex decomposable. Furthermore, we prove that if $I$ is a monomial ideal
of degree $2$ then $I$ is weakly polymatroidal if and only if $I$ has linear
quotients if and only if $I$ is vertex splittable.
