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

03/11/2019

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by   Günter Rote, et al.

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A tree with n vertices has at most 95^n/13 minimal dominating sets. The growth constant λ = √(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.