Criticality of Switching Classes of Reversible 2-Structures Labeled by an Abelian Group

Abstract

Let V be a finite vertex set and let (𝔸, +) be a finite abelian group. An 𝔸-labeled and reversible 2-structure defined on V is a function g : (V × V) \ {(v, v) : v ∈ V } → 𝔸 such that for distinct u, v ∈ V, g(u, v) = −g(v, u). The set of 𝔸-labeled and reversible 2-structures defined on V is denoted by ℒ(V, 𝔸). Given g ∈ ℒ(V, 𝔸), a subset X of V is a clan of g if for any x, y ∈ X and v ∈ V \ X, g(x, v) = g(y, v). For example, ∅, V and {v} (for v ∈ V) are clans of g, called trivial. An element g of ℒ(V, 𝔸) is primitive if |V | ≥ 3 and all the clans of g are trivial.

Keywords

• labeled and reversible 2-structure
• switching class
• clan
• primitivity
• criticality
• 05c75