Edge-Orienting on Split, Planar and Treelike Graphs

Let G(V, E) be an undirected connected graph, where each vertex v is associated with a positive cost C(v) and each edge e 5 (u, v) is associated with two positive weights, W(u!v) and W(v!u). We consider a new graph problem, called the edge-orientation problem (the EOP). The major issue is to assign each edge e 5 (u, v) an orientation, either from u to v, denoted as u!v, or from v to u, denoted as v!u, such that maxx[VfC(x) 1 Sx!z W(x!z)g is minimized. This paper first shows that the EOP is NP-hard on split graphs and planar graphs. Then, a linear-time algorithm on star graphs is proposed by the prune-and-search strategy. Finally, the algorithmic result on star graphs is extended to trees and simple cactus graphs using the dynamic programming strategy