Vertex decomposability and weakly polymatroidal ideals

Let $K$ be a field and $R=K[x_1,\dots ,x_n]$ be the polynomial ring in $n$ variables over a field $K$. Let $\mathrm{\Delta }$ be a simplicial complex on $n$ vertices and $I=I_{\mathrm{\Delta }}$ be its Stanley-Reisner ideal. In this paper, we show that if $I$ is a matroidal ideal then the following conditions are equivalent: $(i)$ $\mathrm{\Delta }$ is sequentially Cohen-Macaulay; $(ii)$ $\mathrm{\Delta }$ is shellable; $(iii)$ $\mathrm{\Delta }$ is vertex decomposable. Also, if $I$ is a minimally generated by $u_1,\dots ,u_s$ such that $s\le 3$ or ${\rm supp}(u_i)\cup {\rm supp}(u_j)={x_1,\ldots,x_n}$forall$i\neq j$,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.