Power Domination In Graphs Research Articles

Hardness Results of Connected Power Domination for Bipartite Graphs and Chordal Graphs

A set [Formula: see text] of a graph [Formula: see text] is called a connected power dominating set of [Formula: see text] if [Formula: see text], the subgraph induced by [Formula: see text], is connected and every vertex in the graph can be observed from [Formula: see text], following the two observation rules for power system monitoring: Rule [Formula: see text]: if [Formula: see text], then [Formula: see text] can observe itself and all its neighbors, and Rule [Formula: see text]: for an already observed vertex whose all neighbors except one are observed, then the only unobserved neighbor becomes observed as well. Given a graph [Formula: see text], Minimum Connected Power Domination is to find a connected power dominating set of minimum cardinality of [Formula: see text] and Decide Connected Power Domination is the decision version of Minimum Connected Power Domination. Decide Connected Power Domination is known to be NP -complete for general graphs. In this paper, we prove that Decide Connected Power Domination remains NP -complete for star-convex bipartite graphs, perfect elimination bipartite graphs and split graphs. This answers some open problems posed in [B. Brimkov, D. Mikesell and L. Smith, Connected power domination in graphs, J. Comb. Optim. 38(1) (2019) 292–315]. On the positive side, we show that Minimum Connected Power Domination is polynomial-time solvable for chain graphs, a proper subclass of perfect elimination bipartite graph, and for threshold graphs, a proper subclass of split graphs. Further, we show that Minimum Connected Power Domination cannot be approximated within [Formula: see text] for any [Formula: see text] unless [Formula: see text], for bipartite graphs as well as for chordal graphs. Finally, we show that Minimum Connected Power Domination is APX -hard for bounded degree graphs.

Power domination in maximal planar graphs

Power domination in graphs emerged from the problem of monitoring an electrical system by placing as few measurement devices in the system as possible. It corresponds to a variant of domination that includes the possibility of propagation. For measurement devices placed on a set S of vertices of a graph G, the set of monitored vertices is initially the set S together with all its neighbors. Then iteratively, whenever some monitored vertex v has a single neighbor u not yet monitored, u gets monitored. A set S is said to be a power dominating set of the graph G if all vertices of G eventually are monitored. The power domination number of a graph is the minimum size of a power dominating set. In this paper, we prove that any maximal planar graph of order n ≥ 6 admits a power dominating set of size at most (n−2)/4 .

Connected power domination in graphs

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.

Restricted power domination and zero forcing problems

Power domination in graphs arises from the problem of monitoring an electric power system by placing as few measurement devices in the system as possible. A power dominating set of a graph is a set of vertices that observes every vertex in the graph, following a set of rules for power system monitoring. A practical problem of interest is to determine the minimum number of additional measurement devices needed to monitor a power network when the network is expanded and the existing devices remain in place. In this paper, we study the problem of finding the smallest power dominating set that contains a given set of vertices X. We also study the related problem of finding the smallest zero forcing set that contains a given set of vertices X. The sizes of such sets in a graph G are respectively called the restricted power domination number and restricted zero forcing number of G subject to X. We derive several tight bounds on the restricted power domination and zero forcing numbers of graphs, and relate them to other graph parameters. We also present exact and algorithmic results for computing the restricted power domination number, including integer programs for general graphs and a linear time algorithm for graphs with bounded treewidth. We also use restricted power domination to obtain a parallel algorithm for finding minimum power dominating sets in trees.

On the $$\varvec{k}$$ k -power domination of hypergraphs

The study of electric power networks led to the definition and study of power domination in graphs. This notion was later extended to k-power domination. In the current paper we start the study of the k-power domination in hypergraphs. In particular, we give a structural characterization of the k-power domination number for hypertrees. We also establish a linear-time algorithm to solve the k-power domination problem in hypertrees. Finally, lower and upper bounds for the k-power domination number of standard and non standard connected (r-uniform) hypergraphs are obtained.

Generalized Power Domination in Regular Graphs

In this paper, we continue the study of power domination in graphs (see [T. W. Haynes et al., SIAM J. Discrete Math., 15 (2002), pp. 519--529; P. Dorbec et al., SIAM J. Discrete Math., 22 (2008), pp. 554--567; A. Aazami et al., SIAM J. Discrete Math., 23 (2009), pp. 1382--1399]). Power domination in graphs was birthed from the problem of monitoring an electric power system by placing as few measurement devices in the system as possible. A set of vertices is defined to be a power dominating set of a graph if every vertex and every edge in the system is monitored by the set following a set of rules (according to Kirschoff laws) for power system monitoring. The minimum cardinality of a power dominating set of a graph is its power domination number. We show that the power domination of a connected cubic graph on $n$ vertices different from $K_{3,3}$ is at most $n/4$ and this bound is tight. More generally, we show that for $k \ge 1$, the $k$-power domination number of a connected $(k+2)$-regular graph on $n$ vertices different from $K_{k+2,k+2}$ is at most $n/(k+3)$, where the $1$-power domination number is the ordinary power domination number. We show that these bounds are tight.

Improved Algorithms and Complexity Results for Power Domination in Graphs

The NP-complete Power Dominating Set problem is an “electric power networks variant” of the classical domination problem in graphs: Given an undirected graph G=(V,E), find a minimum-size set P⊆V such that all vertices in V are “observed” by the vertices in P. Herein, a vertex observes itself and all its neighbors, and if an observed vertex has all but one of its neighbors observed, then the remaining neighbor becomes observed as well. We show that Power Dominating Set can be solved by “bounded-treewidth dynamic programs.” For treewidth being upper-bounded by a constant, we achieve a linear-time algorithm. In particular, we present a simplified linear-time algorithm for Power Dominating Set in trees. Moreover, we simplify and extend several NP-completeness results, particularly showing that Power Dominating Set remains NP-complete for planar graphs, for circle graphs, and for split graphs. Specifically, our improved reductions imply that Power Dominating Set parameterized by |P| is W[2]-hard and it cannot be better approximated than Dominating Set.

Power domination in graphs

The problem of monitoring an electric power system by placing as few measurement devices in the system as possible is closely related to the well-known domination problem in graphs. In 1998, Haynes et al. considered the graph theoretical representation of this problem as a variation of the domination problem. They defined a set S to be a power dominating set of a graph if every vertex and every edge in the system is monitored by the set S (following a set of rules for power system monitoring). The power domination number γ P ( G ) of a graph G is the minimum cardinality of a power dominating set of G . In this paper, we present upper bounds on the power domination number for a connected graph with at least three vertices and a connected claw-free cubic graph in terms of their order. The extremal graphs attaining the upper bounds are also characterized.

A note on power domination in grid graphs

The problem of monitoring an electric power system by placing as few measurement devices in the system as possible is closely related to the well known vertex covering and dominating set problems in graphs (see [T.W. Haynes, S.M. Hedetniemi, S.T. Hedetniemi, M.A. Henning, Power domination in graphs applied to electrical power networks, SIAM J. Discrete Math. 15(4) (2002) 519–529]). A set S of vertices is defined to be a power dominating set of a graph if every vertex and every edge in the system is monitored by the set S (following a set of rules for power system monitoring). The minimum cardinality of a power dominating set of a graph is its power domination number. In this paper, we determine the power domination number of an n × m grid graph.