Generalized Powers of Graphs and Their Algorithmic Use

Motivated by the frequency assignment problem in heterogeneous multihop radio networks, where different radio stations may have different transmission ranges, we introduce two new types of coloring of graphs, which generalize the well-known Distance-k-Coloring. Let G = (V,E) be a graph modeling a radio network, and assume that each vertex v of G has its own transmission radius r(v), a nonnegative integer. We define r-coloring (r-coloring) of G as an assignment Φ : V → {0, 1, 2, . . .} of colors to vertices such that Φ(u) = Φ(v) implies dG(u, v) > r(v) + r(u) (dG(u, v) > r(v) + r(u) + 1, respectively). The r-Coloring problem (the r-Coloring problem) asks for a given graph G and a radius-function r : V → N ∪ {0}, to find an r-coloring (an rcoloring, respectively) of G with minimum number of colors. Using a new notion of generalized powers of graphs, we investigate the complexity of the r-Coloring and r-Coloring problems on several families of graphs.