Minimal Dominating Sets in a Tree: Counting, Enumeration, and Extremal Results

Abstract

A tree with n vertices has at most 95 n/13 minimal dominating sets. The growth constant λ = 13 √ 95 ≈ 1.4194908 is best possible. It is obtained in a semi-automatic way as a kind of "dominant eigenvalue" of a bilinear operation on sixtuples that is derived from the dynamic-programming recursion for computing the number of minimal dominating sets of a tree. We also derive an output-sensitive algorithm for listing all minimal dominating sets with linear set-up time and linear delay between successive solutions.