Power domination throttling

Abstract

A power dominating set of a graph G = (V, E) is a set S ⊂ V that colors every vertex of G according to the following rules: in the first timestep, every vertex in N [S] becomes colored; in each subsequent timestep, every vertex which is the only non-colored neighbor of some colored vertex becomes colored. The power domination throttling number of G is the minimum sum of the size of a power dominating set S and the number of timesteps it takes S to color the graph. In this paper, we determine the complexity of power domination throttling and give some tools for computing and bounding the power domination throttling number. Some of our results apply to very general variants of throttling and to other aspects of power domination.

Throttling for Zero Forcing and Variants

“…Recall that for two graphs G 1 and G 2 , the graph G 1 ∪G 2 has vertex set and edge set equal to V (G 1 ) ∪V (G 2 ) and E(G 1 ) ∪E(G 2 ) respectively. For any graph G, let X(G) be the set of subsets of V (G) that are ⌊Z⌋ forcing sets of G. A list of conditions is given in [3,Theorem 1] that would guarantee that Z Floor Throttling is NP-Complete.…”

“…However, S 1 is not a ⌊Z⌋ forcing set of G 1 . So the conditions given in [3,Theorem 1] cannot be used to prove that Z Floor Throttling is NP-Complete. It would be useful to have other tools to help determine the complexity of the Z Floor Throttling problem.…”

“…Let G ′ be the graph obtained by contracting a path edge of K 3 P 3 shown in the middle of Figure 4. The degree sequences of G ′ and G are (5,4,4,3,3,3,3,3) and (5,3,3,3,3,3,3,3) respectively. Thus, the only possible way to delete edges in G ′ and obtain G is by deleting the edge between the two vertices of degree 4.…”

“…The (standard) throttling number of G is the minimum value of th(G, B) where B ranges over all zero forcing sets of G. For a given graph G and an integer k, the Zero Forcing Throttling problem is to determine if the standard throttling number of G is less than k. The many variations of zero forcing (see [2]) lead to many variations of throttling. In [3], it was shown that Zero Forcing Throttling and other variants are NP-Complete.…”

Optimizing the trade-off between number of cops and capture time in Cops and Robbers

“…The cop throttling number, which optimizes the sum of the resources used to accomplish a task and the time to accomplish the task, follows in the established study of throttling for other parameters (cf. [9,10,11,12]). In the case of Cops and Robbers, arguably it is the person-hours that should be optimized, i.e., the product rather than the sum.…”