Let G = (V, E) be an undirected graph, and P ⊆ V any subset of nodes. By a P -path we will mean a simple (non-self-intersecting on the level of nodes) path with distinct endpoints, both in P. We will identify paths with their sets of edges which we in turn will identify with their characteristic vectors. The characteristic vector of F ⊆ E is simply ∑_{e∈F} e considered as an element in the vector space W having one basis vector for each edge of G.

We think of the graph as a computer network, the nodes being routers and the edges being links. It’s possible to send packets through the network, and we assume that sending a packet through the edge e takes time t_{e} (and is constant). The time needed to send a packet through a sequence of edges is the sum of the t_{e} for each edge e in the path. The nodes in P are special routers capable of sending and receiving packets (the other routers do not have this capacity, they just mindlessly forward the packets). So by sending a packet from a router A ∈ P to another router B ∈ P along a prespecified P -path α, one measures ∑_{e∈α} t_{e}. The following question becomes natural: Question: Given the value ∑_{e∈α} t_{e} for each P -path α, is it possible to compute the individual values t_{e} for each edge e ∈ E?

Clearly this is asking whether the set of vectors {α is a P -path} span W or not.

Theorem 0.1. The P -paths span W if and only if for every pair u, v of vertices (possibly contained in P), every connected component of G−u−v contains some node in P.

We first show that all cycles are contained in the span.

Lemma 0.1. Assuming the hypothesis of the theorem, any cycle in G is contained in the span of the P -paths.

Proof. Let C be any cycle of G. Consider the graph G′ obtained from G by adding a node s adjacent to each node in P and a node t adjacent to each node in C. The graph G′ is 3-connected, thus there are three edge-disjoint paths from s to t. These three paths contain three mutually disjoint paths A, B, C, each starting in P and ending in a, b, c respectively, where a, b, c are distinct nodes in C.

Let α = b\overrightarrow{C}c, β = c\overrightarrow{C}a, γ = a\overrightarrow{C}b be the three parts of C between a, b, c.