Line Graphs of Switching Classes

A signed graph Σ is a graph with signed edges [3]. A switching class (of signed graphs) is an equivalence class of signed graphs under the switching relation: Σ1 ∼ Σ2 ⇐ ⇒ V (Σ1) = V (Σ1) = V, E(Σ1) = E(Σ2), and (∃X ⊆ V) an edge has the same sign in Σ1 and Σ2 iff both ends are in X or both in V \X [7]. A switching class [Σ] is characterized by the underlying graph Γ = (V, E) and the class of positive circuits, B = {C: C is a circuit of Σ whose edge signs have positive product}. The triple 〈Σ 〉 = (V, E, B) is a sign-biased graph; it is equivalent to [Σ]. Our graphs will be finite and loopless. Let 〈Σ 〉 be a sign-biased graph with underlying graph Γ = (V, E) and let VL = E, E ′ L = {{v, e, f} : e, f ∈ E and v is a vertex of e and f}. Then (VL, E ′ L) is the ordinary line graph L(Γ). A circuit in it is called “derived ” (from C0) if its vertices are the edges of a circuit C0 of Γ, a “vertex triangle ” if it is a triangle whose vertices are edges at a common vertex in Γ. Let B ′ L = {C: C is a circuit of (VL, E ′ L) and its edge set is a sum of derived circuits and vertex triangles, all but an even numbe