New algorithms for discrete vector optimization based on the Graef-Younes method and cone-monotone sorting functions

Abstract

The well-known Jahn-Graef-Younes algorithm, proposed by Jahn in 2006, generates all minimal elements of a finite set with respect to an ordering cone. It consists of two Graef-Younes procedures, namely the forward iteration, which eliminates a part of the non-minimal elements, followed by the backward iteration, which is applied to the reduced set generated by the previous iteration. Without using the backward iteration, we develop new algorithms that also compute all minimal elements of the initial set, by combining the forward iteration with certain sorting procedures based on cone-monotone functions. In particular, when the ordering cone is polyhedral, computational results obtained in MATLAB allow us to compare our algorithms with the Jahn-Graef-Younes algorithm, within a bi-objective optimization problem.     

On the Number of k‐Dominating Independent Sets

We study the existence and the number of k‐dominating independent sets in certain graph families. While the case namely the case of maximal independent sets—which is originated from Erd?s and Moser—is widely investigated, much less is known in general. In this paper we settle the question for trees and prove that the maximum number of k‐dominating independent sets in n‐vertex graphs is between and if , moreover the maximum number of 2‐dominating independent sets in n‐vertex graphs is between and . Graph constructions containing a large number of k‐dominating independent sets are coming from product graphs, complete bipartite graphs, and finite geometries. The product graph construction is associated with the number of certain Maximum Distance Separable (MDS) codes.     

5-正则图的全控制数的一个注记

设G是不含孤立点的图,S是G的一个顶点子集,若G的每一个顶点都与S中的某顶点邻接,则称S是G的全控制集.G的最小全控制集所含顶点的个数称为G的全控制数,记为γt(G).Thomasse和Yeo证明了若G是最小度至少为5的n阶连通图,则γt(G)≤17n/44.在5-正则图上改进了Thomasse和Yeo的结论,证明了若G是n阶5-正则图,则,γt(G)≤106n/275.     

NA随机变量阵列的完全矩收敛性

利用NA随机变量的矩不等式和截尾方法,研究了NA随机变量阵列的完全矩收敛性,给出了证明NA随机变量阵列完全矩收敛性的一些充分条件.所得结果推广了已有文献关于NA随机变量的相应结果.     

乘积图的Fractional控制

Let G =(V,E) be a simple graph.For any real function g :V-→ R and a subset S V,we write g(S) =∑v∈Sg(v).A function f :V-→ [0,1] is said to be a fractional dominating function(F DF) of G if f(N [v]) ≥ 1 holds for every vertex v ∈ V(G).The fractional domination number γf(G) of G is defined as γf(G) = min{f(V)|f is an F DF of G }.The fractional total dominating function f is defined just as the fractional dominating function,the difference being that f(N(v)) ≥ 1 instead of f(N [v]) ≥ 1.The fractional total domination number γ0f(G) of G is analogous.In this note we give the exact values ofγf(Cm × Pn) and γ0f(Cm × Pn) for all integers m ≥ 3 and n ≥ 2.     

On the Characterization of Maximal Planar Graphs with a Given Signed Cycle Domination Number

Let G =(V, E) be a simple graph. A function f : E → {+1,-1} is called a signed cycle domination function(SCDF) of G if ∑_(e∈E(C))f(e) ≥ 1 for every induced cycle C of G. The signed cycle domination number of G is defined as γ'_(sc)(G) = min{∑_(e∈E)f(e)| f is an SCDF of G}. This paper will characterize all maximal planar graphs G with order n ≥ 6 and γ'_(sc)(G) = n.