Group based graph transformations and hierarchical representations of graphs

Abstract

A labeled 2-structure, ℓ2s for short, is a complete edgelabeled directed graph without loops or multiple edges. An importaxtt result of the theory of 2-structures is the existence of a hierarchical representation of each ℓ2s. ∆ δ -reversible labeled 2-structure g will be identiffed with its labeling function that maps each edge (z, y), z ≠y, of the domain D into a group ∆ so that g(y, z) = δ(g(x, y)) for an involution b of ∆. For each mapping (selector) σ: D →, ∆ a b-reversible 2-structure g is obtained from g by gσ (z, y) =σ (z)g(z, y)8(σ (y)). A dynamic δ-reversible 2-structure G = [g] generated by g is the set {gσ [σ a selector}. We define the plane trees of G to capture the hierarchical representation of G as seen by individual elements of the domain. We show that all the plane trees axe strongly related to each other. Indeed, they are all obtainable from one simple unrooted undirected tree - the form of G. Thus, quite surprisingly, all hierarchical representations of ℓ2s's belonging to one dynamic ℓ2s G can be combined into one hierarchical representation of G.