Forwarding and optical indices of a graph

Motivated by wavelength-assignment problems for all-to-all traffic in optical networks, we study graph parameters related to sets of paths connecting all pairs of vertices. We consider sets of both undirected and directed paths, under minimisation criteria known as edge congestion and wavelength count; this gives rise to four parameters of a graph GG: its edge forwarding index π(G)π(G), arc forwarding index π→(G), undirected optical index w(G), and directed optical index w→(G).In the paper we address two long-standing open problems: whether the equality π→(G)=w→(G) holds for all graphs, and whether indices π(G)π(G) and w(G) are hard to compute. For the first problem, we give an example of a family of planar graphs {Gk}{Gk} such that π→(Gk)≠w→(Gk). For the second problem, we show that determining either π(G)π(G) or w(G) is NP-hard.