20 research outputs found
Restricted power domination and fault-tolerant power domination on grids
AbstractThe power domination problem is to find a minimum placement of phase measurement units (PMUs) for observing the whole electric power system, which is closely related to the classical domination problem in graphs. For a graph G=(V,E), the power domination number of G is the minimum cardinality of a set S⊆V such that PMUs placed on every vertex of S results in all of V being observed. A vertex with a PMU observes itself and all its neighbors, and if an observed vertex with degree d>1 has only one unobserved neighbor, then the unobserved neighbor becomes observed. Although the power domination problem has been proved to be NP-complete even when restricted to some special classes of graphs, Dorfling and Henning in [M. Dorfling, M.A. Henning, A note on power domination in grid graphs, Discrete Applied Mathematics 154 (2006) 1023–1027] showed that it is easy to determine the power domination number of an n×m grid. Their proof provides an algorithm for giving a minimum placement of PMUs. In this paper, we consider the situation in which PMUs may only be placed within a restricted subset of V. Then, we present algorithms to solve this restricted type of power domination on grids under the conditions that consecutive rows or columns form a forbidden zone. Moreover, we also deal with the fault-tolerant measurement placement in the designed scheme and provide approximation algorithms when the number of faulty PMUs does not exceed 3
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
Distributed Dominating Sets on Grids
This paper presents a distributed algorithm for finding near optimal
dominating sets on grids. The basis for this algorithm is an existing
centralized algorithm that constructs dominating sets on grids. The size of the
dominating set provided by this centralized algorithm is upper-bounded by
for grids and its difference
from the optimal domination number of the grid is upper-bounded by five. Both
the centralized and distributed algorithms are generalized for the -distance
dominating set problem, where all grid vertices are within distance of the
vertices in the dominating set.Comment: 10 pages, 9 figures, accepted in ACC 201
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
On Triangular Secure Domination Number
Let T_m=(V(T_m), E(T_m)) be a triangular grid graph of m ϵ N level. The order of graph T_m is called a triangular number. A subset T of V(T_m) is a dominating set of T_m if for all u_V(T_m)\T, there exists vϵT such that uv ϵ E(T_m), that is, N[T]=V(T_m). A dominating set T of V(T_m) is a secure dominating set of T_m if for each u ϵ V(T_m)\T, there exists v ϵ T such that uv ϵ E(T_m) and the set (T\{u})ꓴ{v} is a dominating set of T_m. The minimum cardinality of a secure dominating set of T_m, denoted by γ_s(T_m) is called a secure domination number of graph T_m. A secure dominating number  γ_s(T_m) of graph T_m is a triangular secure domination number if γ_s(T_m) is a triangular number. In this paper, a combinatorial formula for triangular secure domination number of graph T_m was constructed. Furthermore, the said number was evaluated in relation to perfect numbers