Classification of Indecomposable Flows of Signed Graphs

An indecomposable flow $f$ on a signed graph $\Sigma$ is a nontrivial integral flow that cannot be decomposed into $f=f_1+f_2$, where $f_1,f_2$ are nontrivial integral flows having the same sign (both $\geq 0$ or both $\leq 0$) at each edge of $\Sigma$. This paper is to classify indecomposable flows into characteristic vectors of circuits and Eulerian cycle-trees --- a class of signed graphs having a kind of tree structure in which all cycles can be viewed as vertices of a tree. Moreover, each indecomposable flow other than circuit characteristic vectors can be further decomposed into a sum of certain half circuit characteristic vectors having the same sign at each edge. The variety of indecomposable flows of signed graphs is much richer than that of ordinary unsigned graphs.