Balanced decompositions of a signed graph

The balanced decomposition number (b.d.n.) δ₀(Σ) of a signed graph Σ is the smallest number of balanced subsets into which its edges can be partitioned. (A special case is decomposition of a graph into bipartite subgraphs.) The connected b.d.n. δ₁(Σ) is the same, but the subsets must also be connected. The balanced partition number (b.p.n.) π₀(Σ) is the smallest size of a partition of the vertices into subsets inducing balanced subgraphs; the connected b.p.n. π₁(Σ) is similar but the induced subgraphs must be connected. We use signed graph coloring theory to prove that δ₀ = 1 + ⌈log₂ π₀⌉ and that δ₀ is analogous to a critical exponent in the sense of Crapo and Rota. We deduce bounds on δ₀ and values for special signed graphs. We show that δ₀ is computationally about as complex as the chromatic number. We prove that, for a complete signed graph, δ₁ = δ₀; more strongly, with three exceptions a minimal balanced decomposition exists into connected and spanning edge sets. And we show that, of δ₁ and 1 + ⌈log₂ π₁⌉, neither is always at least as large as the other.