Strongly connected orientations of plane graphs

We study the problem of orienting a subset of edges of a given plane graph such that the resulting sub-digraph is strongly connected and spans all vertices of the graph. We are interested in orientations with minimum number of arcs which at the same time produce a digraph with smallest possible stretch factor. Such orientations have applications into the problem of establishing strongly connected sensor network when sensors are equipped with directional antennae.We present three constructions for such orientations. Let G=(V,E)G=(V,E) be a 2-edge connected plane graph and let Φ(G)Φ(G) be the degree of the largest face in GG. Our constructions are based on a face coloring of GG, say with λλ colors.The first construction gives a strongly connected orientation with at most (2−4λ−6λ(λ−1))|E| arcs and the stretch factor at most Φ(G)−1Φ(G)−1. The second construction gives a strongly connected orientation with |E||E| arcs and the stretch factor at most (Φ(G)−1)⌈λ+12⌉. The third construction can be applied to plane graphs which are 3-edge connected. It uses a particular 6-face coloring and for any integer k≥1k≥1 it produces a strongly connected orientation with at most (1−k10(k+1))|E| arcs and the stretch factor at most Φ2(G)(Φ(G)−1)2k+4Φ2(G)(Φ(G)−1)2k+4. Since the stretch factor solely depends only on Φ(G)Φ(G), λλ, and kk, if these three parameters are bounded, our constructions result in orientations with bounded stretch factor.