In this paper, we propose computational approaches for the zero forcing problem, the connected zero forcing problem, and the problem of forcing a graph within a specified number of timesteps.Our approaches are based on a combination of integer programming models and combinatorial algorithms, and include formulations for zero forcing as a dynamic process, and as a set-covering problem. We explore several solution strategies for these models, test them on various types of graphs, and show that they are competitive with the state-of-the-art algorithm for zero forcing.Our proposed algorithms for connected zero forcing and for controlling the number of zero forcing timesteps are the first general-purpose computational methods for these problems, and are superior to brute force computation.Connected zero forcing is a variant of zero forcing in which the initially colored set of vertices induces a connected subgraph. The connected zero forcing number of a graph is the cardinality of the smallest connected set of initially colored vertices which forces the entire graph to be colored (i.e., the smallest connected zero forcing set). Applications and various structural and computational aspects of connected zero forcing have been investigated in [15,16,17]; in particular, it can be used for modeling the spread of ideas or diseases originating from a single connected source in a network, or for power network monitoring accounting for the cost of supporting infrastructure. Other variants of zero forcing, such as positive semidefinite zero forcing [11,33,44,57], fractional zero forcing, signed zero forcing [41], and k-forcing [5,48] have also been studied. These are typically obtained by modifying the zero forcing color change rule, or adding certain restrictions to a zero forcing set. The number of timesteps in the zero forcing process after which a graph becomes colored is also a problem of interest (see, e.g., [14,21,28,43,57]). Connected variants of other graph problems -such as connected domination and connected power domination [25,31,36,40] -have been extensively studied as well.A closely related problem to zero forcing is power domination, where given a set S of initially colored vertices, the zero forcing color change rule is applied to N [S] instead of to S. Integer programming formulations for power domination and its variants have been explored in [1,18].The power domination problem is derived from the phase measurement unit (PMU) placement problem in electrical engineering, which has also been studied extensively; see, e.g., [47,49] and the bibliographies therein for various integer programming models and combinatorial algorithms for the PMU placement problem. Another closely related problem to zero forcing is the target set selection problem, where given a set S of initially colored vertices and a threshold function θ : V (G) → Z, all uncolored vertices v that have at least θ(v) colored neighbors become colored. Thus, the zero forcing problem constrains the infectors, but the target set selection problem constrai...

We consider the cop-throttling number of a graph G for the game of Cops and Robbers, which is defined to be the minimum of (k + capt k (G)), where k is the number of cops and capt k (G) is the minimum number of rounds needed for k cops to capture the robber on G over all possible games. We provide some tools for bounding the copthrottling number, including showing that the positive semidefinite (PSD) throttling number, a variant of zero forcing throttling, is an upper bound for the cop-throttling number. We also characterize graphs having low cop-throttling number and investigate how large the cop-throttling number can be for a given graph. We consider trees, unicyclic graphs, incidence graphs of finite projective planes (a Meyniel extremal family of graphs), a family of cop-win graphs with maximum capture time, grids, and hypercubes. All the upper bounds on the cop-throttling number we obtain for families of graphs are O( √ n).

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...

The distance matrix $\mathcal{D}(G)$ of a graph $G$ is the matrix containing the pairwise distances between vertices, and the distance Laplacian matrix is $\mathcal{D}^L (G)=T(G)-\mathcal{D} (G)$, where $T(G)$ is the diagonal matrix of row sums of $\mathcal{D}(G)$. Several general methods are established for producing $\mathcal{D}^L$-cospectral graphs that can be used to construct infinite families. Examples are provided to show that various properties are not preserved by $\mathcal{D}^L$-cospectrality, including examples of $\mathcal{D}^L$-cospectral strongly regular and circulant graphs. It is established that the absolute values of coefficients of the distance Laplacian characteristic polynomial are decreasing, i.e., $|\delta^L_{1}|\geq \cdots \geq |\delta^L_{n}|$, where $\delta^L_{k}$ is the coefficient of $x^k$.