Approximations of the Domination Number of a Graph

Let G be a graph with an ordered set of vertices and maximum degree ∆. The domination number γ(G) of G is the minimum order of a set S of vertices such that each vertex not in S is adjacent to some vertex in S. Equivalently, we can label the vertices from {0, 1} so that the sum over each closed neighborhood is at least one; the minimum value of the sum of all labels, with this restriction, is the domination number. The fractional domination number γ∗(G) is defined in the same way except that the vertex labels are chosen from [0, 1]. Let γg(G) be the approximation of the domination number by the standard greedy algorithm. Computing the domination number is NP-complete; however, we can bound γ by these two more easily computed parameters: γ∗(G) ≤ γ(G) ≤ γg(G). How good are these approximations? Using techniques from the theory of hypergraphs, one can show that, for every graph G of order n, γg(G) γ∗(G) = O(log n). On the other hand, we provide examples of graphs for which γ/γ∗ = Θ(log n) and graphs for which γg/γ = Θ(log n). Lastly, we use our examples to compare two bounds on γg . In the following, G will represent a finite, simple, undirected graph. We denote by δ(G) and ∆(G) the minimum and maximum degree of G, respectively. We use N [v] to denote the closed neighborhood of a vertex v. The closed neighborhood of a sequence of vertices, e.g., N [v1, v2, . . . , vk], is the union of the closed neighborhoods of the vertices in the sequence. We denote the domination number of G by γ(G). See [9] for an introduction to domination in graphs and definitions of graph-theoretic terms. We may consider a dominating set as a 0, 1-weighting of the vertex set so that, in each closed neighborhood, the sum of the weights is at least one. Relaxing the requirement that the weights be integers, we obtain a fractional version of the domination number. Suppose we assign weight f(v) ∈ [0, 1] to each vertex v. The function f : V (G) → [0, 1] is a fractional domination if for each vertex v, ∑