Trees with extremal numbers of k-dominating sets
In this paper we investigate the relation between the number of k-dominating sets and the number of independent sets in a tree. We call the k-interior of a tree T the forest that contains all the vertices of T of degree at least k. Heuberger and Wagner (2008) characterized for all n,d≥3 the n-vertex trees Td,n with maximum vertex degree d and the maximum number of independent sets. We show that for all n,k≥2, if a tree T has the maximum number of k-dominating sets among all n-vertex trees, then either it contains exactly 2⌊n−2k−1⌋ k-dominating sets or its k-interior is isomorphic to a forest aK1∪Tk,b for some integers a,b≥0. Moreover, we completely describe trees with the minimum possible number of k-dominating sets for every k≥2.