5 research outputs found
Randomized algorithms and upper bounds for multiple domination in graphs and networks
We consider four different types of multiple domination and provide new improved upper bounds for the k- and k-tuple domination numbers. They generalize two classical bounds for the domination number and are better than a number of known upper bounds for these two multiple domination parameters. Also, we explicitly present and systematize randomized algorithms for finding multiple dominating sets, whose expected orders satisfy new and recent upper bounds. The algorithms for k- and k-tuple dominating sets are of linear time in terms of the number of edges of the input graph, and they can be implemented as local distributed algorithms. Note that the corresponding multiple domination problems are known to be NP-complete. 漏 2011 Elsevier B.V. All rights reserved
Multiple domination models for placement of electric vehicle charging stations in road networks
Electric and hybrid vehicles play an increasing role in the road transport
networks. Despite their advantages, they have a relatively limited cruising
range in comparison to traditional diesel/petrol vehicles, and require
significant battery charging time. We propose to model the facility location
problem of the placement of charging stations in road networks as a multiple
domination problem on reachability graphs. This model takes into consideration
natural assumptions such as a threshold for remaining battery load, and
provides some minimal choice for a travel direction to recharge the battery.
Experimental evaluation and simulations for the proposed facility location
model are presented in the case of real road networks corresponding to the
cities of Boston and Dublin.Comment: 20 pages, 5 figures; Original version from March-April 201
A Note on Triple Repetition Sequence of Domination Number in Graphs
A set D subset of V(G) is a dominating set of a graph G if for all x 系 V(G)\D, for some y 系 D such that xy 系 E(G). A dominating set D subset of V(G) is called a connected dominating set of a graph G if the subgraph \u3cD\u3e induced by D is connected. A connected domination number of G, denoted by 纬_c(G), is the minimum cardinality of a connected dominating set D. The triple repetition sequence denoted by {S_n:n 系 Z+} is a sequence of positive integers which is repeated thrice, i.e., {S_n}={1,1,1,2,2,2,3,3,3, ...}. In this paper, we construct a combinatorial explicit formula for the triple repetition sequence of connected domination numbers of a triangular grid graph
On the approximability and exact algorithms for vector domination and related problems in graphs
We consider two graph optimization problems called vector domination and
total vector domination. In vector domination one seeks a small subset S of
vertices of a graph such that any vertex outside S has a prescribed number of
neighbors in S. In total vector domination, the requirement is extended to all
vertices of the graph. We prove that these problems (and several variants
thereof) cannot be approximated to within a factor of clnn, where c is a
suitable constant and n is the number of the vertices, unless P = NP. We also
show that two natural greedy strategies have approximation factors ln D+O(1),
where D is the maximum degree of the input graph. We also provide exact
polynomial time algorithms for several classes of graphs. Our results extend,
improve, and unify several results previously known in the literature.Comment: In the version published in DAM, weaker lower bounds for vector
domination and total vector domination were stated. Being these problems
generalization of domination and total domination, the lower bounds of 0.2267
ln n and (1-epsilon) ln n clearly hold for both problems, unless P = NP or NP
\subseteq DTIME(n^{O(log log n)}), respectively. The claims are corrected in
the present versio