Bounds on the connected domination number of a graph

Computational approaches for zero forcing and related problems

“…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.…”

“…29) ensures that vertices connected to α cannot be used to connect to any other vertices;(30) and(31) ensure that there are no cycles in the chosen edges. Constraint(32) ensures that vertices chosen to be in the forcing set must be in the spanning tree instead of connected to α.…”

Connected power domination in graphs

“…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 problem called "connected power domination" in [27] is slightly different from the one considered here; see Section 6 for details). The connected variants of other graph problems, including connected zero forcing [13,14,15], connected domination [20,23,29,47], and connected vertex cover [19,36], have also been extensively studied. Imposing connectivity often fundamentally changes the nature of a problem, including its complexity, structural properties, and applications.…”

“…Connected power domination was explored from a computational perspective in [27] (although the problem called "connected power domination" in [27] is slightly different from the one considered here; see Section 6 for details). The connected variants of other graph problems, including connected zero forcing [13,14,15], connected domination [20,23,29,47], and connected vertex cover [19,36], have also been extensively studied. Imposing connectivity often fundamentally changes the nature of a problem, including its complexity, structural properties, and applications.…”

Some comments on the Slater number

“…Simple modifications of the proof of Theorem 1 imply that it is NP-complete to decide whether γ t (G) = sℓ t (G) for a given graph G. Also the above results concerning sparse graphs can be extended to sℓ t (G) and the paired/total domination number. In [2], Desormeaux, Haynes, and Henning define the connected order-sum number ord c (G) of a graph G with non-increasing degree sequence d 1 ≥ . .…”