When I asked this question before I got nary a nibble. That means either the question was too weird to resonate with anyone out there, or I did not make it clear that there was a question there that I did not know the answer to. This will attempt to clarify things, so if I get no nibble on this one, I will chalk it up to weirdness.

My current project led me to construct signed graphs as follows. The resulting graph $\Gamma$ will be a connected, signed, undirected graph with finitely many edges, with parallel edges allowed, and loops allowed, but it will shortly be obvious that loops can be ignored. The construction proceeds by ordering (enumerating) the edges as $e_1$, $e_2$, $\ldots$, $e_n$, and the edges are thought of as being added one by one, starting with the empty graph. It is convenient to let $\Gamma_i$ be the subgraph consisting of the edges $e_1$ through $e_i$ and let $\Gamma_0$ be the graph with no edges. The sign of edge $e_i$ will be positive if and only if its endpoints have valences (degrees) of like parity in $\Gamma_{i-1}$. It is clear that the signs depend on the ordering of the edges.

What is important in the construction is whether the result is balanced (every closed walk passes over an even number of negative edges).

Boundaries of polygons are easy to analyze, and the result depends only on the parity of the number of edges and is independent of the ordering. So far, experiment with pen and paper seems to hint that with every other graph, the outcome depends heavily on the edge ordering and that both balanced and unbalanced can occur. Some take a bit of experimenting to get a balanced outcome. Unbalanced outcomes are easy. Are there other graphs where the outcome is independent of the ordering of the edges?

Does anyone have any insight?