A maximal independent set is an independent set that is not a proper subset of any other independent set. A connected graph (respectively, graph) G with vertex set V(G) is called a quasi-tree graph (respectively, quas...A maximal independent set is an independent set that is not a proper subset of any other independent set. A connected graph (respectively, graph) G with vertex set V(G) is called a quasi-tree graph (respectively, quasi-forest graph), if there exists a vertex x ∈V(G) such that G −x?is a tree (respectively, forest). In this paper, we survey on the large numbers of maximal independent sets among all trees, forests, quasi-trees and quasi-forests. In addition, we further look into the problem of determining the third largest number of maximal independent sets among all quasi-trees and quasi-forests. Extremal graphs achieving these values are also given.展开更多
G. C. Ying, Y. Y. Meng, B. E. Sagan, and V. R. Vatter [1] found the maximum number of maximal independent sets in connected graphs which contain at most two cycles. In this paper, we give an alternative proof to deter...G. C. Ying, Y. Y. Meng, B. E. Sagan, and V. R. Vatter [1] found the maximum number of maximal independent sets in connected graphs which contain at most two cycles. In this paper, we give an alternative proof to determine the largest number of maximal independent sets among all connected graphs of order n ≥ 12, which contain at most two cycles. We also characterize the extremal graph achieving this maximum value.展开更多
The aim of this paper is to given an algebraic computational method for finding maximal independent sets as well as the independent number of an arbitrary finite graph of n vertices G by strengthening the problem of f...The aim of this paper is to given an algebraic computational method for finding maximal independent sets as well as the independent number of an arbitrary finite graph of n vertices G by strengthening the problem of finding maximal independent sets of G to the problem of finding k-independent sets in G for. It is shown that the existence of k-independent sets in G is equivalent to the existence of solutions of a system of multivariate polynomial equations. It follows that the problem of finding k-independent sets can be realized by using Gröbner bases of polynomial ideals. Since the number of k-independent sets is finite, the triangular equations composed by Gröbner bases are easier to be solved. Consequently, the maximal independent sets and the independent number of G are obtained after solving at most n such equations. Finally, the numerical example is presented to illustrate the effectiveness of this algebraic computational method.展开更多
To cope with the constraint problem of power consumption and transmission delay in the virtual backbone of wireless sensor network, a distributed connected dominating set （CDS） algorithm with （α,β）-constraints i...To cope with the constraint problem of power consumption and transmission delay in the virtual backbone of wireless sensor network, a distributed connected dominating set （CDS） algorithm with （α,β）-constraints is proposed. Based on the （α, β）-tree concept, a new connected dominating tree with bounded transmission delay problem（CDTT） is defined and a corresponding algorithm is designed to construct a CDT-tree which can trade off limited total power and bounded transmission delay from source to destination nodes. The CDT algorithm consists of two phases： The first phase constructs a maximum independent set（MIS）in a unit disk graph model. The second phase estimates the distance and calculates the transmission power to construct a spanning tree in an undirected graph with different weights for MST and SPF, respectively. The theoretical analysis and simulation results show that the CDT algorithm gives a correct solution to the CDTF problem and forms a virtual backbone with（ α,β）-constraints balancing the requirements of power consumption and transmission delay.展开更多
由于无线传感器网络节点的能量具有不可再生性,为了减小和均衡网络中各节点的能量损耗,要求把能效高放在首位,以尽可能的延长网络生存期。文中介绍一种利用图论中极大独立集和极小支配集的概念设计的基于能量的有网关的多级簇树EAMCT-G(...由于无线传感器网络节点的能量具有不可再生性,为了减小和均衡网络中各节点的能量损耗,要求把能效高放在首位,以尽可能的延长网络生存期。文中介绍一种利用图论中极大独立集和极小支配集的概念设计的基于能量的有网关的多级簇树EAMCT-G(Energy-Aware Multilevel Clustering Tree with Gateway)算法,并提出该算法的局部维护和更新算法,使得EAMCT-G算法具有可扩展性好和自恢复能力,最后通过仿真验证算法的有效性。展开更多
对规模较大、移动较频繁的MANET(Mobile Ad hoc Networks),用独立支配集构建虚拟骨干网,克服骨干节点之间必须维护连通性的问题,使得拓扑变化较快时骨干网的重构能快速实现;利用极大独立集的求解得到极小独立支配集,并给出基于该支配集...对规模较大、移动较频繁的MANET(Mobile Ad hoc Networks),用独立支配集构建虚拟骨干网,克服骨干节点之间必须维护连通性的问题,使得拓扑变化较快时骨干网的重构能快速实现;利用极大独立集的求解得到极小独立支配集,并给出基于该支配集的虚拟骨干网数学模型及算法;通过仿真验证算法的有效性、低复杂度和自恢复能力.展开更多
文摘A maximal independent set is an independent set that is not a proper subset of any other independent set. A connected graph (respectively, graph) G with vertex set V(G) is called a quasi-tree graph (respectively, quasi-forest graph), if there exists a vertex x ∈V(G) such that G −x?is a tree (respectively, forest). In this paper, we survey on the large numbers of maximal independent sets among all trees, forests, quasi-trees and quasi-forests. In addition, we further look into the problem of determining the third largest number of maximal independent sets among all quasi-trees and quasi-forests. Extremal graphs achieving these values are also given.
文摘G. C. Ying, Y. Y. Meng, B. E. Sagan, and V. R. Vatter [1] found the maximum number of maximal independent sets in connected graphs which contain at most two cycles. In this paper, we give an alternative proof to determine the largest number of maximal independent sets among all connected graphs of order n ≥ 12, which contain at most two cycles. We also characterize the extremal graph achieving this maximum value.
文摘The aim of this paper is to given an algebraic computational method for finding maximal independent sets as well as the independent number of an arbitrary finite graph of n vertices G by strengthening the problem of finding maximal independent sets of G to the problem of finding k-independent sets in G for. It is shown that the existence of k-independent sets in G is equivalent to the existence of solutions of a system of multivariate polynomial equations. It follows that the problem of finding k-independent sets can be realized by using Gröbner bases of polynomial ideals. Since the number of k-independent sets is finite, the triangular equations composed by Gröbner bases are easier to be solved. Consequently, the maximal independent sets and the independent number of G are obtained after solving at most n such equations. Finally, the numerical example is presented to illustrate the effectiveness of this algebraic computational method.
基金Major Program of the National Natural Science Foundation of China (No.70533050)High Technology Research Program ofJiangsu Province(No.BG2007012)+1 种基金China Postdoctoral Science Foundation(No.20070411065)Science Foundation of China University of Mining andTechnology(No.OC080303)
文摘To cope with the constraint problem of power consumption and transmission delay in the virtual backbone of wireless sensor network, a distributed connected dominating set （CDS） algorithm with （α,β）-constraints is proposed. Based on the （α, β）-tree concept, a new connected dominating tree with bounded transmission delay problem（CDTT） is defined and a corresponding algorithm is designed to construct a CDT-tree which can trade off limited total power and bounded transmission delay from source to destination nodes. The CDT algorithm consists of two phases： The first phase constructs a maximum independent set（MIS）in a unit disk graph model. The second phase estimates the distance and calculates the transmission power to construct a spanning tree in an undirected graph with different weights for MST and SPF, respectively. The theoretical analysis and simulation results show that the CDT algorithm gives a correct solution to the CDTF problem and forms a virtual backbone with（ α,β）-constraints balancing the requirements of power consumption and transmission delay.
文摘由于无线传感器网络节点的能量具有不可再生性,为了减小和均衡网络中各节点的能量损耗,要求把能效高放在首位,以尽可能的延长网络生存期。文中介绍一种利用图论中极大独立集和极小支配集的概念设计的基于能量的有网关的多级簇树EAMCT-G(Energy-Aware Multilevel Clustering Tree with Gateway)算法,并提出该算法的局部维护和更新算法,使得EAMCT-G算法具有可扩展性好和自恢复能力,最后通过仿真验证算法的有效性。
文摘对规模较大、移动较频繁的MANET(Mobile Ad hoc Networks),用独立支配集构建虚拟骨干网,克服骨干节点之间必须维护连通性的问题,使得拓扑变化较快时骨干网的重构能快速实现;利用极大独立集的求解得到极小独立支配集,并给出基于该支配集的虚拟骨干网数学模型及算法;通过仿真验证算法的有效性、低复杂度和自恢复能力.
