Sequentially Sr simplicial complexes and sequentially S 2 graphs

Hassan Haghighi, Naoki Terai, Siamak Yassemi, Rahim Zaare-Nahandi

Research output: Contribution to journalArticlepeer-review

11 Citations (Scopus)


We introduce sequentially Sr modules over a commutative graded ring and sequentially Sr simplicial complexes. This generalizes two properties for modules and simplicial complexes: being sequentially Cohen-Macaulay, and satisfying Serre's condition Sr. In analogy with the sequentially Cohen-Macaulay property, we show that a simplicial complex is sequentially Sr if and only if its pure i-skeleton is Sr 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 Sr cycles showing that the only sequentially S2 cycles are odd cycles and, for r ≥ 3, no cycle is sequentially Sr 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 Sr 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.

Original languageEnglish
Pages (from-to)1993-2005
Number of pages13
JournalProceedings of the American Mathematical Society
Issue number6
Publication statusPublished - Jun 2011
Externally publishedYes


  • Sequentially Cohen-Macaualy
  • Sequentially Sr simplicial complex
  • Serre's condition

ASJC Scopus subject areas

  • Mathematics(all)
  • Applied Mathematics


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