Superextensions and the depth of median graphs

An invariant of convex structures—the depth—is used to study the structure of finite median graphs. The main result is a recursive description of graphs of given depth. This leads to a complete description of the cubical structure of the superextensionλ(5) (answering a question in [19]) and to a (less complete) description of superextensionsλ(r) forr > 5. An important tool is the construction of certain graphsϱ(r) of linked bipartitions of ther-point setr. For eachr ⩾ 3, the graphsλ(r) and ϱ(r) have the same number of vertices, but they are isomorphic (modulo extreme points) forr ⩽ 5 only.

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