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. © 1987.
Zasĺavsky, T. (1987). Balanced decompositions of a signed graph. Journal of Combinatorial Theory, Series B, 43(1), 1–13. https://doi.org/10.1016/0095-8956(87)90026-8