Vertex Dismissibility and Scalability of Simplicial Complexe

Abstract

We study three nonpure extensions of vertex decomposability, shellability, and Cohen-Macaulayness at the minimum facet dimension. For $b=\operatorname{indim}\Delta$, strong vertex dismissibility gives a deletion-link recursion on the complex itself in which deletion does not decrease $b$, while vertex dismissibility and scalability require the pure initial skeleton $\Delta^{[b]}$ to be vertex decomposable and shellable, respectively. We prove that a join is strongly vertex dismissible exactly when both factors are strongly vertex dismissible. We also show that strong vertex dismissibility implies that every skeleton $\Delta^k=\Delta^{[k]}$, $k\le b$, is vertex decomposable. When $\operatorname{indim}\Delta\leq1$, weak connectedness is equivalent to vertex dismissibility, scalability, and initially Cohen-Macaulayness over some, equivalently every, field. For quasi-forests, these conditions are also equivalent to strong vertex dismissibility. Our main graph result classifies independence complexes of pseudoforests by showing that strong vertex dismissibility, vertex dismissibility, scalability, and initially Cohen-Macaulayness over some or every field are equivalent and are characterized by a componentwise independent domination criterion. Finally, we connect the combinatorial properties of the pure initial skeleton with algebraic properties of its Alexander dual, focusing on vertex splittability, linear quotients, and linear resolutions in fixed degree.