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We introduce a graph decomposition which exists for all simple, connectedgraphs $G=(V,E)$. The decomposition $V = A \cup B \cup C$ is such that eachvertex in $A$ has more neighbors in $B$ than in $A$ and vice versa. $C$ is`balanced': each $v \in C$ has the same number of neighbours in $A$ and $B$.These decompositions arise naturally from the behavior of an associateddynamical system (`Randomized Rounding') on $(\mathbb{S}^1)^{|V|}$. Connectionsto judicious partitions and the \textsc{MaxCut} problem (in particular theBurer-Monteiro-Zhang heuristic) are being discussed.

A Graph Decomposition motivated by the Geometry of Randomized Rounding