6 research outputs found
Heredity for generalized power domination
In this paper, we study the behaviour of the generalized power domination
number of a graph by small changes on the graph, namely edge and vertex
deletion and edge contraction. We prove optimal bounds for
, and for in
terms of , and give examples for which these bounds are
tight. We characterize all graphs for which for any edge . We also consider the behaviour of the
propagation radius of graphs by similar modifications.Comment: Discrete Mathematics and Theoretical Computer Science, 201
Power Domination Number On Shackle Operation with Points as Lingkage
The Power dominating set is a minimum point of determination in a graph that can dominate the connected dots around it, with a minimum domination point. The smallest cardinality of a power dominating set is called a power domination number with the notation . The purpose of this study is to determine the Shackle operations graph value from several special graphs with a point as a link. The result operation graphs are: Shackle operation graph from Path graph , Shackle operation graph from Sikel graph , Shackle operation graph from Star graph . The method used in this paper is axiomatic deductive method in solving problems. Understanding the axiomatic method itself is a method of deductive proof principles that applies in mathematical logic by using theorems that already exist in solving a problem. In this paper begins by determining the paper object that is the Shackle point operations result graph. Next, determine the cardinality of these graphs. After that, determine the point that has the maximum degree on the graph as the dominator point of power domination. Then, check whether the nearest neighbor has two or more degrees and analyze its optimization by using a ceiling function comparison between zero forching with the greatest degree of graph. Thus it can be determined ϒp minimal and dominated. The results of the power domination number study on Shackle operation graph result with points as connectors are , for  and ; , for  and ; , for  and
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
Nordhaus–Gaddum problems for power domination
A power dominating set of a graph G is a set S of vertices that can observe the entire graph under the rules that (1) the closed neighborhood of every vertex in S is observed, and (2) if a vertex and all but one of its neighbors are observed, then the remaining neighbor is observed; the second rule is applied iteratively. The power domination number of G, denoted by gamma p(G), is the minimum number of vertices in a power dominating set. A Nordhaus-Gaddum problem for power domination is to determine a tight lower or upper bound on gamma p(G) + gamma p(G) or gamma p(G).gamma p(G), where G denotes the complement of G. The upper and lower Nordhaus-Gaddum bounds over all graphs for the power domination number follow from known bounds on the domination number and examples. In this note we improve the upper sum bound for the power domination number substantially for graphs having the property that both the graph and its complement are connected. For these graphs, our bound is tight and is also significantly better than the corresponding bound for the domination number. We also improve the product upper bound for the power domination number for graphs with certain properties
The relationship between k-forcing and k-power domination
Zero forcing and power domination are iterative processes on graphs where an initial set of vertices are observed, and additional vertices become observed based on some rules. In both cases, the goal is to eventually observe the entire graph using the fewest number of initial vertices. The concept of k-power domination was introduced by Chang et al. (2012) as a generalization of power domination and standard graph domination. Independently, k-forcing was defined by Amos et al. (2015) to generalize zero forcing. In this paper, we combine the study of k-forcing and k-power domination, providing a new approach to analyze both processes. We give a relationship between the k-forcing and the k-power domination numbers of a graph that bounds one in terms of the other. We also obtain results using the contraction of subgraphs that allow the parallel computation of k-forcing and k-power dominating sets
Generalized power domination in regular graphs
International audienceIn this paper, we continue the study of power domination in graphs (see SIAM J. Discrete Math. 15 (2002), 519-529; SIAM J. Discrete Math. 22 (2008), 554-567; SIAM J. Discrete Math. 23 (2009), 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 K3,3 is at most n/4 and this bound is tight. More generally, we show that for k ≥ 1 the k-power domination number of a connected (k + 2)-regular graph on n vertices different from Kk+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