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If $J\subset I$ are two monomials ideals, we give a practical upper bound forthe Stanley depth of $J/I$, which we call it the \emph{quasi-depth} of $J/I$.Also, we compute the quasi-depth of several classes of square free monomialideals. We also study the Stanley depth of the monomial ideal associated of theindependent sets of a graph.

On the quasi-depth of squarefree monomial ideals and the sdepth of the monomial ideal of independent sets of a graph