\begin{abstract} Let $K$ be a field and $R = K[x_1,\ldots,x_n]$ be the polynomial ring in $n$ variables over $K$. 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: \begin{enumerate} \item $\Delta$ is sequentially Cohen--Macaulay; \item $\Delta$ is shellable; \item $\Delta$ is vertex decomposable. \end{enumerate} We also prove that if $I$ is minimally generated by $u_1,\ldots,u_s$ such that $s \leq 3$ or \[ \operatorname{supp}(u_i)\cup \operatorname{supp}(u_j) =\{x_1,\ldots,x_n\} \] for all $i\neq j$, then $\Delta$ is vertex decomposable. Furthermore, we show that if $I$ is a monomial ideal generated in degree $2$, then the following conditions are equivalent: \begin{enumerate} \item $I$ is weakly polymatroidal; \item $I$ has linear quotients; \item $I$ is vertex splittable. \end{enumerate} \end{abstract}
