The Canonical Vertex Signature and the Cosets of the Complete Binary Cycle Space

We consider two combinatorially simple alternatives to summation with even-degree edge sets for characterizing the class of edge sets E in Kn that have specified odd vertices: replacing a path inside E by a path outside it, or summing with a circle contained in E or in Ec. The latter is equivalent to summation with even-degree edge sets for almost all n, and the former is not quite similarly equivalent. The results help to understand the canonical vertex signature of a signed graph. One way to transform a graph into another is to take the set sum of its edge set with that of a circle (circuit, cycle) contained in the graph or in its complement. (Call this operation circle replacement.) It is well known that a graph with all even degrees is an edge-disjoint union of circles; thus, it can be transformed into any other even-degree graph (on the same vertices) by repeated circle replacement. If we apply the same operation to a graph with some odd-degree vertices, can we get any other graph with the same odd vertices? The answer is that we can—but as is so often true in combinatorial problems, with just a few exceptions. This transformation rule is a restriction of set summation with an arbitrary circle (circle addition), which is known to generate precisely all edge sets with the same odd vertices. The restriction is that we may use only permitted circles, and permission is determined by how the circle interacts with the given edge set E. The restriction in circle replacement is that the circle must lie within E or its complement. A somewhat similar restriction is to permit only a circle made up of two paths, one in E and the other outside it; call this circular path shifting. With circular path shifting we can get almost, but not quite any other edge set with the same odd vertices; there are now infinitely many exceptions which, fortunately, can be described exactly. Signed graphs. I asked this question the better to understand the canonical vertex signature of a signed graph. A signed graph Σ = (Γ, σ) consists of an underlying graph Γ (all our graphs are simple and undirected) and an edge signature σ : E → {+,−}. Quite some time ago E. Sampathkumar introduced the idea of marking the vertices with signs derived