A study on r-configurations - A resource assignment problem on graphs

Let G be an undirected graph with a set of vertices V and a set of edges E. Given an integer r, we assign at most r labels (representing "resources") to each vertex. We say that such an assignment is an r-configuration if, for each label c, the vertices labeled by c form a dominating set for G. In this paper, we are interested in the maximum number, D_r(G), of labels that can be assigned to the vertices of graph G by an r-configuration. The decision problem "D₁(G) ≥ K?" (known as the domatic number problem) is NP-complete. We first investigate D_r(G) for general graphs and establish bounds on D_r(G) in terms of D₁(G) and the minimum vertex degree of G. We then discuss D_r(G) for d-regular graphs. We clearly have D_r(G) ≤ r(d + 1). We show that the problem of testing if D_r(G) = r(d + 1) is solvable in polynomial time for d-regular graphs with |V| = 2(d + 1), but is NP-complete for those with \V\ = a(d + 1) for some integer a ≥ 3. Finally, we discuss cubic (i.e., 3-regular) graphs. It is easy to show 2r ≤ D_r(G) ≤ 4r for cubic graphs. We show that the decision problem for D₁(G) = K is co-NP-complete for K = 2 and is NP-complete for K = 4. Although there are many cubic graphs G with D₁(G) = 2, surprisingly, every cubic graph has a 2-configuration with five labels, i.e., D₂(G) ≥ 5 and such a 2-configuration can be constructed in polynomial time. We use this fact to show D_r(G) ≥ ⌊5r/2⌋ in general.