Submodular decompositions: a bounded-variation counterexample and an exact planar cut formula

Bérczi, Gehér, Imolay, Lovász, and Schwarcz (SIAM J. Discrete Math. 40(2026), 308--335) asked whether every set function of bounded variation is a difference of two bounded submodular functions, and how to determine the optimum in a monotonic sum-decomposition of a planar graph cut function. In this paper, we answer both questions and obtain quantitative extensions. For the first question, we characterize the continuous profiles $f:[0,1]\to\R$ for which $A\mapsto f(m(A))$ is a bounded submodular difference on an atomless probability space. More precisely, we express the minimum oscillation of the subtracted component through the positive part of the distributional second derivative. We also characterize admissible discontinuous profiles and give an exact summability criterion for countable block functions. In particular, the first question has a negative answer even for increasing functions of variation one whose scalar profiles have log-Lipschitz derivatives and belong to every $C^{1,α}$, $0<α<1$. Turning to graph cuts, we prove that the optimum equals the total edge weight minus the maximum fractional triangle-packing weight for every nonnegative weighting if and only if the graph is $K_5$-free. Moreover, maximum packings parametrize the symmetrized optimal components, while on $K_4$-free graphs we describe all optimal components and characterize uniqueness. Consequently, the formula applies to weighted planar graphs and implies edge-weight monotonicity on $K_5$-free graphs. For rational weights, it also yields polynomial-time constructions.