Vertex cover ideals of simplicial complexes

Given a simplicial complex $\mathrm{\Delta }$, we investigate how to construct a new simplicial complex $\bar{\mathrm{\Delta }}$ such that the corresponding monomial ideals satisfy nice algebraic properties. We give a procedure to check the vertex decomposability of an arbitrary hypergraph. As a consequence, we prove that attaching non-pure skeletons at all vertices of a cycle cover of a simplicial complex $\mathrm{\Delta }$ results in a simplicial complex $\bar{\mathrm{\Delta }}$ such that the associated hypergraph $\mathcal{H}(\bar{\mathrm{\Delta }})$ is vertex decomposable. Also, we prove that all symbolic powers of the cover ideal of $\bar{\mathrm{\Delta }}$ are componentwise linear. Our work generalizes the earlier known result where non-pure complete graphs were added to all vertices of a cycle cover of a graph.