Resolution of indecomposable integral flows on signed graphs

It is well-known that each nonnegative integral flow of a directed graph can be decomposed into a sum of nonnegative graph circuit flows, which cannot be further decomposed into nonnegative integral sub-flows. This is equivalent to saying that indecomposable flows of graphs are those graph circuit flows. Turning from graphs to signed graphs, the indecomposable flows are much richer than that of ordinary unsigned graphs. The present paper is to give a complete description of indecomposable flows of signed graphs from the viewpoint of resolution of singularities by introducing covering graphs. A real flow (also known as circulation) on a graph or a signed graph (a graph with signed edges) is a real-valued function on oriented edges such that the net inflow to each vertex is zero. An integral flow is a flow whose values are integers. There are many reasons to be interested in flows on graphs; an important one is their relationship to graph structure through the analysis of (conformally) indecomposable flows, that is, integral flows that cannot be decomposed as the sum of two integral flows having the same sign on each edge (both ≥ 0 or both≤ 0). It is well known, and an important observation in the theory of integral network flows, that the indecomposable flows are identical to the circuit flows, which have value 1 on the edges of a graph circuit (=cycle) and 0 on all other edges. Extending the theory of indecomposable integral flows to signed graphs, which is studied in [7] by algorithmic method, led to a remarkable discovery that there are, besides the anticipated circuit flows (which are already more complicated in signed graphs than in ordinary graphs), many “strange” indecomposable flows with elaborate structure not describable by signed-graph circuits. In this article we characterize this structure again by the method of sign-labeled covering graphs, lifting each vertex and each edge of a signed graph to two vertices and two edges respectively of a sign-labeled covering graph, and lifting each indecomposable flow to a simple cycle flow in the sign-labeled covering graph. We think of the lifting as a combinatorial analog of resolution of singularities in algebraic geometry. The strange indecomposable flows are singular phenomena, which we resolve by lifting them (blowing up overlapped edges) to ordinary cycle flows in a covering graph. Comparing to the algorithmic approach in [7], the present paper hints a connection (at least conceptually) between graph theory and covering spaces of algebraic topology and resolution of algebraic geometry. We believe that this connection is useful to study gain graphs [14] that are more complicated than signed graphs. Date: January 29, 2014. 2000 Mathematics Subject Classification. Primary 05C22; Secondary 05C20, 05C21, 05C38.