Vertex Decomposable Simplicial Complexes Associated to Path Graphs

Characterizing simplicial complexes with vertex decomposability property is one of the main problems in combinatorial commutative algebra since they have many interesting algebraic and topological properties. In this regard introducing families of simplicial complexes with this property is of great interest and has been studied in many research papers. In this paper the Stanley-Reisner simplicial complex associated to the t-clique ideal of the complement of path graphs has been studied. For such a simplicial complex   we explain the set of facets of  and using this characterization we show that every such simplicial complex is vertex decomposable, whose shedding vertex is an endpoint of the path graph. Indeed this family of simplicial complexes are Cohen-Macaulay, since they are pure. Since edge ideals of graphs are in fact 2-clique ideals, this family of  simplicial complexes contains the independence complexes of complement of path graphs. Finally, as a consequence it is shown that the t-independence ideal of the complement of a path graph is vertex splittable and its Betti splitting is presented.