2-Rainbow Domination and Its Practical Variation on Weighted Graphs

Abstract

Let G(V, E) be a simple graph with n-vertex-set V and m-edge-set E. Two positive weights, w ₁ (v) and w ₂ (v), are assigned to each vertex v. For each v ∈ V, let N(v) = {u | u ∈ V and (u, v) ∈ E} and N[v] = {v} ∪ N(v). A 2-rainbow function of G is a function f mapping each vertex v to f(v) = f ₂(v) f ₁(v), f ₂(v), f ₁(v) ∈ {0, 1}. The weight of f is defined as $w(f) = \sum_{v \in V[f_1(v)w_1(v) + f_2(v)w_2(v)]}$. A 2-rainbow function f of G is called a 2-rainbow dominating function if Θ _{u ∈ N(v)} f(u) = 11, for all vertices v with f(v) = 00, where Θ _{u ∈ N(v)} f(u) is the result of performing bit-wise Boolean OR on f(u), for all u ∈ N(v). Our problem is to obtain a 2-rainbow dominating function f of G such that w(f) is minimized. This paper first proves that the problem is NP-hard on unweighted planar graphs. Then, an O(n)-time algorithm for the problem on trees is proposed using the dynamic programming strategy. Finally, a practical variation, called the weighted minimum tuple 2-rainbow domination problem, is proposed and the relationship between it and the weighted minimum domination problem is established.