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We introduce sequentially $S_r$ modules over a commutative graded ring andsequentially $S_r$ simplicial complexes. This generalizes two properties formodules and simplicial complexes: being sequentially Cohen-Macaulay, andsatisfying Serre's condition $S_r$. In analogy with the sequentiallyCohen-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 $r=2$, weprovide a more relaxed characterization. As an algebraic criterion, we provethat a simplicial complex is sequentially $S_r$ if and only if the minimal freeresolution of the ideal of its Alexander dual is componentwise linear in thefirst $r$ steps. We apply these results for a graph, i.e., for the simplicialcomplex of the independent sets of vertices of a graph. We characterizesequentially $S_r$ cycles showing that the only sequentially $S_2$ cycles areodd cycles and, for $r\ge 3$, no cycle is sequentially $S_r$ with the exceptionof cycles of length 3 and 5. We extend certain known results on sequentiallyCohen-Macaulay graphs to the case of sequentially $S_r$ graphs. We prove that abipartite 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 thatany graph with no chordless even cycle is sequentially $S_2$. Finally, wepropose some questions.

Sequentially $S_r$ simplicial complexes and sequentially $S_2$ graphs