Simplicial complexes with rigid depth

We extend a result of Minh and Trung to get criteria for $\depth I=\depth\sqrt{I}$ where $I$ is an unmixed monomial ideal of the polynomial ring $S=K[x_1,..., x_n]$. As an application we characterize all the pure simplicial complexes $\Delta$ which have rigid depth, that is, which satisfy the condition that for every unmixed monomial ideal $I\subset S$ with $\sqrt{I}=I_\Delta$ one has $\depth(I)=\depth(I_\Delta).$