  We introduce sequentially $S_r$ modules over a commutative graded ring and
sequentially $S_r$ simplicial complexes. This generalizes two properties for
modules and simplicial complexes: being sequentially Cohen-Macaulay, and
satisfying Serre's condition $S_r$. In analogy with the sequentially
Cohen-Macaulay property, we show that a simplicial complex is sequentially
$S_r$ if and only if its pure $i$-skeleton is $S_r$ for all $i$. For $r=2$, we
provide a more relaxed characterization. As an algebraic criterion, we prove
that a simplicial complex is sequentially $S_r$ if and only if the minimal free
resolution of the ideal of its Alexander dual is componentwise linear in the
first $r$ steps. We apply these results for a graph, i.e., for the simplicial
complex of the independent sets of vertices of a graph. We characterize
sequentially $S_r$ cycles showing that the only sequentially $S_2$ cycles are
odd cycles and, for $r\ge 3$, no cycle is sequentially $S_r$ with the exception
of cycles of length 3 and 5. We extend certain known results on sequentially
Cohen-Macaulay graphs to the case of sequentially $S_r$ graphs. We prove that a
bipartite graph is vertex decomposable if and only if it is sequentially $S_2$.
We provide some more results on certain graphs which in particular implies that
any graph with no chordless even cycle is sequentially $S_2$. Finally, we
propose some questions.
