I have a graph called partition graph. This graph gives rise to a simplicial complex called box complex $B_{edge}$. Since this simplicial complex is too big and studying the topological features of it is too difficult I want to simplify this simplicial complex using a discrete Morse function.
Definition. Every vertex of a partition graph $\mathcal{P}(3^3)$ is a partition of $\{1,2, ..., 9 \}$ into $3$ cells of size $3$. Two vertices $u$ and $v$ are adjacent if the intersection of each cell of $u$ with each cell of $v$ is nonempty.
This graph is vertex- edge- arc transitive.
Definition. The box complex $B_{edge}(G)$ of a graph $G$ is a simplicial complex which vertices are the directed edges of $G$; that is, ordered pairs $(u,v)$ with $\{u,v\}\in E$. The simplices of this box complex are subsets of edge sets of complete bipartite subgraphs of $G$, where the edges are oriented from the first shore to the second shore. \begin{align*} B_{edge}(G):=&\{ \overrightarrow{F} \subset A'\times A'': \varnothing \neq A',A'' \subset V,\\ &A'\cap A''=\varnothing, G[A',A'']\ \text{is complete}\} \end{align*} I want to find a discrete Morse function on the box complex $B_{edge}(\mathcal{P}(3^3))$. A function with less critical simplicies is better.