The study of power domination in graphs arises from the problem of placing a minimum number of measurement devices in an electrical network while monitoring the entire network. A power dominating set of a graph is a set of vertices from which every vertex in the graph can be observed, following a set of rules for power system monitoring. In this paper, we study the problem of finding a minimum power dominating set which is connected; the cardinality of such a set is called the connected power domination number of the graph. We show that the connected power domination number of a graph is NP-hard to compute in general, but can be computed in linear time in cactus graphs and block graphs. We also give various structural results about connected power domination, including a cut vertex decomposition and a characterization of the effects of various vertex and edge operations on the connected power domination number. Finally, we present novel integer programming formulations for power domination, connected power domination, and power propagation time, and give computational results.vertices; the physical laws by which PMUs can observe a network give rise to the following color change rules (see [17]): 1) Every neighbor of an initially colored vertex becomes colored.2) Whenever there is a colored vertex with exactly one uncolored neighbor, that neighbor becomes colored.S is a power dominating set of G if all vertices in G become colored after applying rule 1) once, and rule 2) as many times as possible (i.e. until no more vertices can change color). The power domination number of G, denoted γ P (G), is the cardinality of a minimum power dominating set. S is a zero forcing set of G if all vertices in G become colored after applying rule 2) as many times as possible (and not applying rule 1) at all). The zero forcing number of G, denoted Z(G), is the cardinality of a minimum zero forcing set. The process of zero forcing was introduced independently in combinatorial matrix theory [5] and in quantum control theory [18].In this paper, we study a variant of power domination which requires every set of initially colored vertices to induce a connected subgraph. Given a connected graph G = (V, E), a set S ⊂ V is a connected power dominating set of G if S is a power dominating set and G[S] is connected. The connected power domination number, denoted γ P,c (G), is the cardinality of a minimum connected power dominating set. Requiring a power dominating set to be connected is motivated by the application in monitoring electrical networks: the data from PMUs is relayed by high-speed communication infrastructure to processing stations which collect and manage this data; thus, in addition to minimizing the production costs of the PMUs, an electric power company may seek to place all PMUs in a compact, connected region in the network in order to reduce the number of processing stations and related infrastructure required to collect the data.Connected power domination was explored from a computational perspective in [27] (although the p...

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.

We present an integer programming model to compute the strong rainbow connection number, src(G), of any simple graph G. We introduce several enhancements to the proposed model, including a fast heuristic, and a variable elimination scheme. Moreover, we present a novel lower bound for src(G) which may be of independent research interest. We solve the integer program both directly and using an alternative method based on iterative lower bound improvement, the latter of which we show to be highly effective in practice. To our knowledge, these are the first computational methods for the strong rainbow connection problem. We demonstrate the efficacy of our methods by computing the strong rainbow connection numbers of graphs containing up to 379 vertices.

We consider the problem of computing the strong rainbow connection number src(G) for cactus graphs G in which all cycles have odd length. We present a formula to calculate src(G) for such odd cacti which can be evaluated in linear time, as well as an algorithm for computing the corresponding optimal strong rainbow edge coloring, with polynomial worst case run time complexity. Although computing src(G) is NP-hard in general, previous work has demonstrated that it may be computed in polynomial time for certain classes of graphs, including cycles, trees and block clique graphs. This work extends the class of graphs for which src(G) may be computed in polynomial time.