Conformal decomposition of integral flows on signed graphs with outer-edges (Q2053688)

Signed graph was introduced by \textit{F. Harary} [Mich. Math. J. 2, 143--146 (1954; Zbl 0056.42103)] to establish a balance theory to motivate social networks. Elementary notions such as circuits and bonds on signed parallel ordinary graphs are developed by \textit{T. Zaslavsky} [Discrete Appl. Math. 4, 47--74 (1982; Zbl 0476.05080)]. A central observation, pointed out by Zaslavsky is the existence of the signed graph matroid, built up initially from a list of ad hoc patterns serving for circuits by checking the matroid axiom. This paper contains basics as well as foundations. A nonzero integral flow \(f\) is conformally decomposable if \(f = f_{1}+ f_{2}\), where \(f_1 ; f_2\) are nonzero integral flows, both are nonnegative or both are nonpositive. Conformally indecomposable flows on ordinary graphs are simply graph circuit flows. However, on signed graphs without outer-edges (compact case), conformally indecomposable flows, classified by \textit{B. Chen} and \textit{J. Wang} [``Classification of indecomposable flows of signed graphs'', Preprint, \url{arXiv:1112.0642}] and by \textit{B. Chen} et al. [Discrete Math. 340, No. 6, 1271--1286 (2017; Zbl 1369.05098)], are signed-graph circuit flows plus an extra class of characteristic flows of so-called Eulerian circle-trees. Here, the author focused to the classification of the indecomposable conformal integral flows on signed graphs with outer-edges (non-compact case). A notable feature is that with outer-edges the treatment is natural and results become stronger but proofs are simpler than that without outeredges. This paper is interesting and useful to researchers in the field of signed graph. Those who are working in this area must look into the paper to get some useful motivation.