Efficient (j, k)-Dominating Functions

For positive integers and , an ( of a graph = ( ) is a function : → {0, 1, 2, . . ., } such that the sum of function values in the closed neighbourhood of every vertex equals . The relationship between the existence of efficient ( )-dominating functions and various kinds of efficient dominating sets is explored. It is shown that if a strongly chordal graph has an efficient ( )-dominating function, then it has an efficient dominating set. Further, every efficient ( )-dominating function of a strongly chordal graph can be expressed as a sum of characteristic functions of efficient dominating sets. For there are strongly chordal graphs with an efficient dominating set but no efficient ( )-dominating function. The problem of deciding whether a given graph has an efficient ( )-dominating function is shown to be NP-complete for all positive integers and , and solvable in polynomial time for strongly chordal graphs when = . By taking = 1 we obtain NP-completeness of the problem of deciding whether a given graph has an efficient -tuple dominating set for any fixed positive integer . Finally, we consider efficient (2, 2)-dominating functions of trees. We describe a new constructive characterization of the trees with an efficient dominating set and a constructive characterization of the trees with two different efficient dominating sets. A number of open problems and questions are stated throughout the work.