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...

In this paper, we consider the problem of sensing a finite set of (moving) objects over a finite planning horizon using a set of sensors in prefixed locations that vary with respect to time over a discretized space. Control in this situation is limited and the problem considered is one of opportunistic sensing. We formulate an integer program that maximizes the quality of sensor return given either deterministic or probabilistic (i.e., forecasted) object routes. We examine the computational complexity of the problem and show it is non-deterministic polynomial-hard. We theoretically and numerically illustrate subclasses of the problem that are computationally simpler, ultimately deriving a heuristic that is strongly polynomial. Real-world and constructed data sets are used in our analysis.

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The minimum k-core problem asks for the smallest induced subgraph of minimum degree k. It has been shown that this problem is NP-hard, and thus sophisticated techniques are required to obtain good solutions and approximations. In this article, the minimum k-core problem is modeled as a binary integer program and relaxed as a linear program. Since the relaxation may yield a non-integral solution, a branch-and-cut framework is used to find an integral optimal solution. It is shown that the edge and cycle transversals of the graph give valid inequalities for the convex hull of the k-core polytope-which can be further generalized to a family of 𝓁-core transversals. Further, a heuristic for the transversal of the minimal 𝓁-cores is given with its associated valid inequality. Additionally, improved valid inequalities are generated using bounds involving the girth of the graph. Multiple heuristics are explored for finding initial bounds for the branching process utilizing the degree distribution of the graph. Finally, numerical results are given comparing the branch-and-bound, branch-and-cut, and heuristic techniques.