Capacitated Domination

We define an r-capacitated dominating set of a graph G = (V,E) as a set {v1, . . . , vk} ⊆ V such that there is a partition (V1, . . . , Vk) of V where for all i, vi ∈ Vi, vi is adjacent to all of Vi − {vi}, and |Vi| ≤ r + 1. kr(G) is the minimum cardinality of an r-capacitated dominating set. We show properties of kr, especially as regards the trivial lower bound |V |/(r+1). We calculate the value of the parameter in several graph families, and show that it is related to codes and polyominoes. The parameter is NP-complete in general to compute, but a greedy approach provides a linear-time algorithm for trees.