# isDecomposable -- determines whether a simplicial complex is k-decomposable

## Synopsis

• Usage:
isDecomposable(k, S)
• Inputs:
• Outputs:
• B, , true if and only if $S$ is $k$-decomposable

## Description

Definition 3.6 of [Wo] states that a simplicial complex $S$ is $k$-decomposable if $S$ is either a simplex or there exists a shedding face $F$ of $S$ of dimension at most $k$ such that both the face deletion and link of $S$ by $F$ are again $k$-decomposable.

 i1 : R = QQ[a..f]; i2 : isDecomposable(0, simplicialComplex {a*b*c*d*e*f}) o2 = true i3 : isDecomposable(2, simplicialComplex {a*b*c, b*c*d, c*d*e}) o3 = true

The method checks the cache, if possible, to see if the complex is vertex-decomposable.

• faceDelete -- computes the face deletion for a simplicial complex
• isSheddingFace -- determines whether a face of a simplicial complex is a shedding face
• isShellable -- determines whether a simplicial complex is shellable
• isVertexDecomposable -- determines whether a simplicial complex is vertex-decomposable