Resolution of Irreducible Integral Flows on a Signed Graph

We completely describe the structure of irreducible integral flows on a signed graph by lifting them to the signed double covering graph. A (real-valued) flow (sometimes also called a circulation) on a graph or a signed graph (a graph with signed edges) is a real-valued function on oriented edges, f : ~ E → R, such that the net inflow to any 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 irreducible flows, that is, integral flows that cannot be decomposed as the sum of other flows of lesser value. It is well known, and an important observation in the thoery of integral network flows, that the irreducible flows are identical to the circuit flows, which have value 1 on the edges of a graph circuit (that is, a cycle) and 0 on all other edges. Extending the theory of irreducible integral flows to signed graphs, which was one of the topics of the doctoral dissertation of Wang [4], led to the remarkable discovery that there are, besides the anticipated circuit flows (which in signed graphs are already more complicated than in unsigned graphs), also many ‘strange’ irreducible flows with elaborate structure not describable by circuits. In this article we characterize that structure by lifting it to a simple cycle in the signed covering graph. (Indeed, this was how we discovered the correct characterization, though we were also guided by the partial result in Wang’s thesis.) We like to think of lifting as a combinatorial analog of resolution of singularities in continuous mathematics. The strange irreducible flows are singular phenomena, which we resolve by lifting them to ordinary cycle flows in a covering graph. This is not a precise statement but a philosophy that we believe will be fruitful. 1. Graphs and signed graphs Graphs. A graph is (V,E), with vertex set V and edge set E. There may be loops and multiple edges. An edge e with endpoints v and w has two ends, which we symbolize by (v, e) and (w, e). A tricky technical point is that this notation does not distinguish the two ends of a loop; we take an easy way out by treating (v, e) and (w, e) as different ends even when v = w. (There are more technically correct means of distinguishing the ends but they make the notation very complicated.) A walk is a sequence W = v0e1v1e2 · · · elvl of vertices and edges such that the endpoints of ei are vi−1 and vi. A walk is closed if l > 0 and v0 = vl and open otherwise. A segment of W is a consecutive subwalk, i.e., viei+1 · · · ejvj. When W is closed we allow j > l, interpreting indices modulo l; thus, a segment may pass through v0. A circle is the edge set of a simple closed walk, i.e., there is no repeated edge or vertex other than that v0 = vl. A graph circuit Date: June 23, 2007. 2000 Mathematics Subject Classification. Primary 05C22; Secondary . The first author’s research was supported by ???