Advanced Mathematical Models & Applications

Let G be a simple graph with the vertex set V(G) = {1,...,n} and edge set E(G). In this paper, we prove that $G$ is vertex decomposable if it does not contain an induced subgraph isomorphic to 2K₂, C₄ or C₅. As a consequence, we show that Stanley’s conjecture holds for the Stanley-Reisner ring of the independence complex of G. We also extend these results to split graphs and chordal graphs (since that the independence complex of a chordal graph is part of a broader class of shellable complexes) often resulting in their Stanley-Reisner rings having nice properties, such as being Cohen-Macaulay or having a linear resolution), showing that their independence complexes are sequentially Cohen–Macaulay since are graded modules over a ring whose filtration by dimension-specific submodules allows each piece to be Cohen–Macaulay, extending the standard Cohen–Macaulay property. They are characterized by their local cohomology, module filtration, or specific Hilbert function properties.). In particular, forests provide an explicit class where Stanley’s conjecture holds. Corollaries are given to show these extensions.