Good edge-labelling of graphs

A good edge-labelling of a graph G is a labelling of its edges such that for any two distinct vertices u, v, there is at most one (u,v)-path with non-decreasing labels. This notion was introduced in [J.-C. Bermond, M. Cosnard, and S. Pérennes. Directed acyclic graphs with unique path property. Technical Report RR-6932, INRIA, May 2009] to solve wavelength assignment problems for specific categories of graphs. In this paper, we aim at characterizing the class of graphs that admit a good edge-labelling. First, we exhibit infinite families of graphs for which no such edge-labelling can be found. We then show that deciding if a graph admits a good edge-labelling is NP-complete. Finally, we give large classes of graphs admitting a good edge-labelling: C3-free outerplanar graphs, planar graphs of girth at least 6, subcubic {C3,K2,3}-free graphs.