Constant Time Computation of Minimum Dominating Sets

Let G be a graph and let P (n) denote an element from a one-parameter family of graphs, such as a path of length n, a cycle of length n, or a complete binary tree of height n. We are concerned with determining minimum dominating sets of graphs of the form G P (n). Using dynamic programming and properties of finite state spaces, we show a constant time algorithm to produce a minimum dominating set of G P (n), for fixed G and all n, for the one-parameter families mentioned. Previous researchers had used similar techniques but obtained only linear-time algorithms. We also show how a closed form expression can be obtained for the minimum domination number of G P (n). We discuss extensions of the algorithm to the determination of all minimum dominating sets for G P (n), and to related problems of coverings, packings, and codes. In addition, we discuss algorithm extensions to several different types of domination, including perfect domination, and to other ways of composing graphs.