A graph with m edges is called label-connected if the edges can be labeled with real numbers in such a way that, for every pair (u, v) of vertices, there is a (u, v)-path with ascending labels. The minimum number of edges of a label-connected graph on n vertices equals the minimum number of calls in the gossip problem for n persons, which is known to be 2n − 4 for n ≥ 4. A polynomial characterization of label-connected graphs with n vertices and 2n − 4 edges is obtained. For a graph G, let θ(G) denote the minimum number of edges that have to be added to E(G) in order to create a graph with two edge-disjoint spanning trees. It is shown that for a graph G to be label-connected, θ(G) ≤ 2 is necessary and θ(G) ≤ 1 is sufficient. For i = 1, 2, the condition θ(G) ≤ i can be checked in polynomial time. Yet recognizing label-connected graphs is an NP-complete problem. This is established by first showing that the following problem is NP-complete: Given a graph G and two vertices u and v of G, does there exist a (u, v)-path P in G such that G−E(P) is connected?