Sequentially S_r simplicial complexes and sequentially S ₂ graphs

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 S₂ cycles are odd cycles and, for r ≥ 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 S₂. We provide some more results on certain graphs which in particular implies that any graph with no chordless even cycle is sequentially S₂. Finally, we propose some questions.