Approximations of the domination number of a graph

Abstract

Given a graph G, the domination number gamma(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, 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 gamma_f(G) is defined in the same way, except that the vertex labels are chosen from [0, 1]. Given an ordering of the vertex set of G, let gamma_g(G) be the approximation of the domination number by the standard greedy algorithm. Computing the domination number is NP-complete; however, we can bound gamma by these two more easily computed parameters: gamma_f(G) <= gamma(G) <= gamma_g(G). How good are these approximations? Using techniques from the theory of hypergraphs, one can show that, for every graph G of order n, gamma_g(G) / gamma_f(G) = O(log n). On the other hand, we provide examples of graphs for which gamma / gamma_f = Theta(log n) and graphs for which gamma_g / gamma = Theta(log n). Lastly, we use our examples to compare two bounds on gamma_g.