MINIMUM CONNECTED r-HOP k-DOMINATING SET IN WIRELESS NETWORKS

2009 ◽  
Vol 01 (01) ◽  
pp. 45-57 ◽  
Author(s):  
DEYING LI ◽  
LIN LIU ◽  
HUIQIANG YANG
Keyword(s):  
Wireless Networks ◽  
Undirected Graph ◽  
Dominating Set ◽  
Approximation Ratio ◽  
Dominating Set Problem ◽  
Simulation Results

In this paper, we study the connected r-hop k-dominating set problem in wireless networks. We propose two algorithms for the problem. We prove that algorithm I for UDG has (2r + 1)3 approximate ratio for k ≤ (2r + 1)2 and (2r + 1)((2r + 1)2 + 1)-approximate ratio for k > (2r + 1)2. And algorithm II for any undirected graph has (2r + 1) ln (Δr) approximation ratio, where Δr is the largest cardinality among all r-hop neighborhoods in the network. The simulation results show that our algorithms are efficient.

scholarly journals Hardness of Mine Sweeper Problem

2018 ◽  
Author(s):  
Thinh D. Nguyen
Keyword(s):  
Natural Number ◽  
Undirected Graph ◽  
Dominating Set ◽  
Dominating Set Problem

Our problem \textsc{Mine Sweeper} is: given an undirected graph where some vertices have a natural number associated with them, decide whether there is a way to mark some of the non-numbered vertices such that the number in each numbered vertex is equal to the amount of its marked neighbours. We reduce \textsc{Dominating Set} problem to ours. \textsc{Dominating Set} is shown to be hard by \cite{GJ}.

scholarly journals On the star forest polytope for trees and cycles

2019 ◽  
Vol 53 (5) ◽  
pp. 1763-1773
Author(s):  
Meziane Aider ◽  
Lamia Aoudia ◽  
Mourad Baïou ◽  
A. Ridha Mahjoub ◽  
Viet Hung Nguyen
Keyword(s):  
Undirected Graph ◽  
Dominating Set ◽  
Discrete Math ◽  
Complete Characterization ◽  
Facial Structure ◽  
Maximum Weight ◽  
Single Node ◽  
End Node ◽  
Vertex Disjoint

Let G = (V, E) be an undirected graph where the edges in E have non-negative weights. A star in G is either a single node of G or a subgraph of G where all the edges share one common end-node. A star forest is a collection of vertex-disjoint stars in G. The weight of a star forest is the sum of the weights of its edges. This paper deals with the problem of finding a Maximum Weight Spanning Star Forest (MWSFP) in G. This problem is NP-hard but can be solved in polynomial time when G is a cactus [Nguyen, Discrete Math. Algorithms App. 7 (2015) 1550018]. In this paper, we present a polyhedral investigation of the MWSFP. More precisely, we study the facial structure of the star forest polytope, denoted by SFP(G), which is the convex hull of the incidence vectors of the star forests of G. First, we prove several basic properties of SFP(G) and propose an integer programming formulation for MWSFP. Then, we give a class of facet-defining inequalities, called M-tree inequalities, for SFP(G). We show that for the case when G is a tree, the M-tree and the nonnegativity inequalities give a complete characterization of SFP(G). Finally, based on the description of the dominating set polytope on cycles given by Bouchakour et al. [Eur. J. Combin. 29 (2008) 652–661], we give a complete linear description of SFP(G) when G is a cycle.

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