Minors of simplicial complexes

Minors of simplicial complexes

We extend the notion of a minor from matroids to simplicial complexes. We show that the class of matroids, as well as the class of independence complexes, is characterized by a single forbidden minor. Inspired by a recent result of Aharoni and Berger, we investigate possible ways to extend the matroid intersection theorem to simplicial complexes.

On Topological Minors in Random Simplicial Complexes

Simplicial Complexes. A (finite abstract) simplicial complex is a finite set system that is closed under taking subsets, i.e., F ⊂ H ∈ X implies F ∈ X. The sets F ∈ X are called faces of X. The dimension of a face F is dim(F ) = |F | − 1. The dimension of X is the maximal dimension of any face. A k-dimensional simplicial complex will also be called a k-complex.

Simplicial Minors and Decompositions of Graphs

Simplicial Decompositions, Tree-decompositions and Graph Minors

The concepts of simplicial decompositions, tree-decompositions and simplicial tree-decompositions were all inspired by a common forerunner: the decompositions of finite graphs used by K. Wagner in his classic paper [ 13 ], in which he proved the equivalence of the 4-Colour-Conjecture to Hadwiger’s Conjecture for n = 5. To show that the 4CC implies Hadwiger’s Conjecture (for n = 5), Wagner used ...

New methods for constructing shellable simplicial complexes

A clutter $mathcal{C}$ with vertex set $[n]$ is an antichain of subsets of $[n]$, called circuits, covering all vertices. The clutter is $d$-uniform if all of its circuits have the same cardinality $d$. If $mathbb{K}$ is a field, then there is a one-to-one correspondence between clutters on $V$ and square-free monomial ideals in $mathbb{K}[x_1,ldots,x_n]$ as follows: To each clutter $mathcal{C}...