Flag graph

A flag graph is a way to express the structure of a polytope's flags.

The flag graph of an abstract polytope is a maniplex, although not all maniplexes are the flag graph of a polytope.

Definition
Two flags of a polytope are $j$-adjacent if they differ in exactly one element of rank $j$.

The flag graph is an edge-colored graph with vertices corresponding to the flags of a polytope and with $i$ colored edges between flags that are $i$-adjacent.

Building the incidence poset
It is trivial to build the flag graph of a ranked poset. The definition is essentially instructions for doing so.


 * Determine the number of flags and create a vertex for each flag.
 * For each rank $i$ determine which vertices are $i$-adjacent and add an edge between them.

However building the poset from the flag graph is not trivial. In fact in general it is not possible as multiple posets can give the same flag graph. We can, however, construct the incidence posets given that we are restricting ourselves to a certain subset of well behaved posets. Specifically if we have a flag graph of an abstract polytope or the compound of abstract polytopes we can perfectly reconstruct the polytope.

Elements


To build a poset we first want to find its elements. Elements of rank $i$ in a a complex correspond to the connected components in the graph with edges of color $i$ removed. This is because when we remove $i$-adjacency from the flag graph any two flags which are connected must pass through the same $i$ element.

To complete the poset we add a maximum and minimum element.

This assumes that every proper section of the polytope is connected. That is either it is a compound or it is flag connected. This is because while it is true that with $i$-edges removed from the flag graph, two connected flags must pass through the same element, it is not necessarily true that flags which are not connected must pass through the same element. Polytopes which are not connected have disconnected flag graphs, and thus if a section is not connected this procedure will determine that the two components of the element are in fact different elements. We assume the maximum number of elements.

Incidence
Two elements are incident on each other if their connected components share a vertex.

Properties

 * Every vertex in a polytope's flag graph has exactly one edge of every color. This is an equivalent statement of dyadicity.
 * Let $P$ be the flag graph of an abstract polytope, for $i$ and $j$ which differ by more than one, the subgraph of $P$ comprised of edges which are colored $i$ or $j$ consists entirely of disconnected cycles of length 4.
 * The flag graph of an abstract polytope is the skeleton of a colorful polytope.