NEIGHBORHOOD UNION OF INDEPENDENT SETS AND HAMILTONICITY OF CLAW-FREE GRAPHS

Let G be a graph,for any u∈V(G),let N(u) denote the neighborhood of u and d(u)=|N(u)| be the degree of u.For any UV(G),let N(U)=∪_~u∈U N(u), and d(U)=|N(U)|.A graph G is called claw-free if it has no induced subgraph isomorphic to K_~1,3 .One of the fundamental results concerning cycles in claw-free graphs is due to Tian Feng,et al.: Let G be a 2-connected claw-free graph of order n,and d(u)+d(v)+d(w)≥n-2 for every independent vertex set {u,v,w} of G, then G is Hamiltonian. It is proved that,for any three positive integers s,t and w,such that if G is a (s+t+w-1)-connected claw-free graph of order n,and d(S)+d(T)+d(W)>n-(s+t+w) for every three disjoint independent vertex sets S,T,W with |S|=s,|T|=t,|W|=w,and S∪T∪W is also independent,then G is Hamiltonian.Other related results are obtained too.

The Number of Maximal Independent Sets in Quasi-Tree Graphs and Quasi-Forest Graphs

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.

Solving the k-Independent Sets Problem of Graphs by Gröbner Bases

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.

An Alternative Proof of the Largest Number of Maximal Independent Sets in Connected Graphs Having at Most Two Cycles

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.

A maximum-independent-set-based channel allocation algorithm for multi-channel wireless networks

A channel allocation algorithm based on the maximum independent set is proposed to decrease network conflict and improve network performance. First, a channel allocation model is formulated and a series of the maximum independent sets （MISs） are obtained from a contention graph by the proposed approximation algorithm with low complexity. Then, a weighted contention graph is obtained using the number of contention vertices between two MISs as a weighted value. Links are allocated to channels by the weighted contention graph to minimize conflicts between independent sets. Finally, after channel allocation, each node allocates network interface cards （NICs） to links that are allocated channels according to the queue lengths of NICs. Simulations are conducted to evaluate the proposed algorithm. The results show that the proposed algorithm significantly improves the network throughput and decreases the end to end delay.

INDEPENDENT-SET-DELETABLE FACTOR-CRITICAL POWER GRAPHS

It is said that a graph G is independent-set-deletable factor-critical （in short, ID-factor-critical）, if, for everyindependent-set I which has the same parity as ｜V（G）｜, G - I has a perfect matching. A graph G is strongly IM-extendable, if for every spanning supergraph H of G, every induced matching of H is included in a perfect matching of H. The κ-th power of G, denoted by G^κ, is the graph with vertex set V（G） in which two vertices are adjacent if and only if they have distance at most k in G. ID-factor-criticality and IM-extendability of power graphs are discussed in this article. The author shows that, if G is a connected graph, then G^3 and T（G） （the total graph of G） are ID-factor-critical, and G^4 （when ｜V（G）｜ is even） is strongly IM-extendable; if G is 2-connected, then D^2 is ID-factor-critical.

A Modified Genetic Algorithm for Maximum Independent Set Problems

genetic algorithm is proposed for maximum independent set problems. A specially designed mutation operato is adopted to search the solution space more efficienily, where adjacen relation of a graph is inte-grated. The DIMACS benchmark graphs are used to test our algorithm, and the results show that the algorithm outper-forms our previous version. Moreover two new low bounds are found for graphs in DIMACS.

Solving the independent set problem by sticker based DNA computers

In this paper, the sticker based DNA computing was used for solving the independent set problem. At first, solution space was constructed by using appropriate DNA memory complexes. We defined a new operation called “divide” and applied it in construction of solution space. Then, by application of a sticker based parallel algorithm using biological operations, independent set problem was resolved in polynomial time.

The Study on the (L,M)-Fuzzy Independent Set Systems

Independent sets play an important role in matroid theory. In this paper, the definitions of pre-independent fuzzy set system and independent fuzzy set system in L-fuzzy setting are presented. Independent M-fuzzifying set system is introduced and some of its properties are discussed. Further independent (L,M)-fuzzy set system is given and some of its properties are obtained. The relations of these independent set systems in the setting of fuzzy vector spaces and fuzzy graphs are showed.

Research on the CPA Audit Independence Risk Assessment Based on the Rough Set Theory

The assessment mode of the CPA audit independence risk based on the Rough Set Theory is a risk assessment method and using the Analytical Hierarchy Process, whose aim is to solve the bid management in the process of the Financial Statement Insurance. Firstly, according to the general instance of the accountant office, the experts grade the risk elements to establish the decision-making table; secondly, construct the judgment matrix using the attribution dependence degree of Variable Precision Rough Set to obtain relative importance, and further get the general importance for all of risk elements; Finally, establish the general assessment mode of the audit independence risk.

基于加权分治技术的set packing精确算法

加权分治技术是算法分析中的一种新技术,该技术基于选择不同的量来描述分支子问题的大小,以求得到在最糟糕情况下最好的时间复杂度.setpacking问题是一典型的NP-hard问题,广泛应用于调度、代码优化和生物信息学等领域.本文对有n个子集的setpacking问题,引入符号全集变量N设计基于分支搜索策略的递归算法,并应用加权分治技术对算法加以分析,得到时间复杂度为O*(1.1686n+N)的精确算法,当N≤n/4时,比现有最佳的算法O*(1.2209n)更加有效.

(α,β)-constraints connected dominating set algorithm in wireless sensor network

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.

A POLYNOMIAL ALGORITHM FOR FINDING THEMINIMUM FEEDBACK VERTEX SET OF A3-REGULAR SIMPLE GRAPH

A subset of the vertex set of a graph is a feedback vertex set of the graph if the resulting graph is a forest after removed the vertex subset from the graph. A polynomial algorithm for finding a minimum feedback vertex set of a 3-regular simple graph is provided.

An Extension of the Win Theorem: Counting the Number of Maximum Independent Sets

Win proved a well-known result that the graph G of connectivity κ(G) withα(G) ≤κ(G) + k-1(k ≥ 2) has a spanning k-ended tree, i.e., a spanning tree with at most k leaves. In this paper, the authors extended the Win theorem in case when κ(G) = 1 to the following: Let G be a simple connected graph of order large enough such that α(G) ≤ k + 1(k ≥ 3) and such that the number of maximum independent sets of cardinality k + 1 is at most n-2k-2. Then G has a spanning k-ended tree.

Maximum Independent Sets and Supervised Learning

The paper discusses an enhancement to a recently presented supervised learning algorithm to solve the Maximum Independent Set problem.In particular,it is shown that the algorithm can be improved by simplifying the task learnt by the neural network adopted,with measurable effects on the quality of the solutions provided on unseen instances.Empirical results are presented to validate the idea..

Nonseparating Independent Sets and Maximum Genus of Graphs

A subset I of vertices of an undirected connected graph G is a nonseparating independent set(NSIS)if no two vertices of I are adjacent and GI is connected.Let Z(G)denote the cardinality of a maximum NSIS of G.A nonseparating independent set containing Z(G)vertices is called the maximum nonseparating independent set.In this paper,we firstly give an upper bound for Z(G)of regular graphs and determine Z(G)for some types of circular graphs.Secondly,we show a relationship between Z(G)and the maximum genus M(G)of a general graph.Finally,an important formula is provided to compute Z(G),i.e.,Z(G)=Σx∈I dI(x)+2(M(G-I)-γM(G))+(ξ(G-I)-ξ(G));where I is the maximum nonseparating independent set and ξ(G)is the Betti deficiency(Xuong,1979)of G.

Independence Numbers in Trees

The independence number of a graph G is the maximum cardinality among all independent sets of G. For any tree T of order n ≥ 2, it is easy to see that . In addition, if there are duplicated leaves in a tree, then these duplicated leaves are all lying in every maximum independent set. In this paper, we will show that if T is a tree of order n ≥ 4 without duplicated leaves, then . Moreover, we constructively characterize the extremal trees T of order n ≥ 4, which are without duplicated leaves, achieving these upper bounds.

Approximation Algorithms for Graph Partition into Bounded Independent Sets

The partition problem of a given graph into three independent sets of minimizing the maximum one is studied in this paper.This problem is NP-hard,even restricted to bipartite graphs.First,a simple 3/2-approximation algorithm for any 2-colorable graph is presented.An improved 7/5-approximation algorithm is then designed for a tree.The theoretical proof of the improved algorithm performance ratio is constructive,thus providing an explicit partition approach for each case according to the cardinality of two color classes.

Path Factors and Neighborhoods of Independent Sets in Graphs

A path-factor is a spanning subgraph F of G such that every component of F is a path with at least two vertices.Let k≥2 be an integer.A P_(≥k)-factor of G means a path factor in which each component is a path with at least k vertices.A graph G is a P_(≥k)-factor covered graph if for any e∈E(G),G has a P_(≥k)-factor including e.Letβbe a real number with 1/3≤β≤1 and k be a positive integer.We verify that(ⅰ)a k-connected graph G of order n with n≥5k+2 has a P_(≥3)-factor if|NG(I)|>β(n-3k-1)+k for every independent set I of G with|I|=「β(2k+1)」;(ⅱ)a(k+1)-connected graph G of order n with n≥5k+2 is a P_(≥3)-factor covered graph if|NG(I)|>β(n-3k-1)+k+1 for every independent set I of G with|I|=「β(2k+1)」.