A Class of Vertex Decomposable Flag Simplicial Complexes

Abstract

In this paper, we investigate the vertex decomposability of clique complexes associated with simple graphs and establishes a structural characterization based on forbidden induced subgraphs. We prove that the clique complex CL(G) is vertex decomposable if and only if the underlying graph G contains no induced subgraphs isomorphic to 2K₂, C₄, or C₅. The proof proceeds by demonstrating that such graphs and their complements are chordal, implying that their independence complexes are vertex decomposable. Since the independence complex of the complement graph coincides with the clique complex of G, the desired result follows. Furthermore, important algebraic consequences are derived: every vertex decomposable complex is shellable and hence partitionable, implying the validity of Stanley’s conjecture for the corresponding face ring K[CL(G)]. Thus, this work introduces a new class of vertex decomposable flag simplicial complexes arising from graphs free of 2K₂, C₄, and C₅. The results provide a significant combinatorial framework connecting chordal graph theory, simplicial complex decomposability, and algebraic properties of Stanley-Reisner rings.