树的四类控制参数的束缚数

图的束缚数是图的控制数研究中的一个重要方面，它在某种程度上反映了图的控制数对边数的敏感度．本文通过对图的结构特征的分析．研究了树的四类控制参数的束缚数，即控制数，强控制数，弱控制数．分数控制数的束缚数．分别给出了其紧的上界．     

关于图的弱符号控制数的下界

图G的弱符号控制数γws（G）有着许多重要的应用背景,因而确定其下界有重要意义.在构造适当点集的基础上,给出了图的弱符号控制数的4个独立的下界,并给出了达到这4个下界的图.     

树的3-彩虹控制数的上界

对树的3-彩虹控制数进行研究,首先用构造法找到直径较小的树的3-彩虹控制数的上界.再通过分类讨论思想和数学归纳法得到一般的阶n大于等于5的树的3-彩虹控制数的上界.     

Signed and minus domination in bipartite graphs

The paper studies the signed domination number and the minus domination number of the complete bipartite graph K _{p, q} .     

控制数与星独立数

本文，我们讨论星独立数、分数星独立数、分数控制数和控制数间的关系，利用规划理论给出了上述参数的一些性质。     

Remarks on restrained domination and total restrained domination in graphs

The restrained domination number ^r(G) and the total restrained domination number _t ^r (G) of a graph G were introduced recently by various authors as certain variants of the domination number (G) of (G). A well-known numerical invariant of a graph is the domatic number d(G) which is in a certain way related (and may be called dual) to (G). The paper tries to define analogous concepts also for the restrained domination and the total restrained domination and discusses the sense of such new definitions.This research was supported by Grant MSM 245100303 of the Ministry of Education, Youth and Sports of the Czech Republic.     

关于Sumner-Blitch猜想的一个注记

设G是一个图。G的最小度，连通度，控制数，独立控制数和独立数分别用δ，k，γ，i和α表示，图G是3－γ-临界的，如果γ=3，而且G增加任一条边所得的图的控制数为2.Sumner和Blitch猜想：任意连通的3－γ临界图满足i=3，本文证明了如果G是使α＝k 1≤δ的连通3－γ-临界图，那么Sumner-Blitch猜想成立。     

Total outer-connected domination numbers of trees

Let G=(V,E) be a graph without an isolated vertex. A set DV(G) is a total dominating set if D is dominating, and the induced subgraph G[D] does not contain an isolated vertex. The total domination number of G is the minimum cardinality of a total dominating set of G. A set DV(G) is a total outer-connected dominating set if D is total dominating, and the induced subgraph G[V(G)−D] is a connected graph. The total outer-connected domination number of G is the minimum cardinality of a total outer-connected dominating set of G. We characterize trees with equal total domination and total outer-connected domination numbers. We give a lower bound for the total outer-connected domination number of trees and we characterize the extremal trees.     

偶阶完全图的点-边全符号控制数

设γ_s~*(G)表示图G的点-边全符号控制数,本文给出了偶阶完全图的点-边全符号控制数的精确值.     

The Twin Domination Numb er of Strong Pro duct of Digraphs

Let γ*(D) denote the twin domination number of digraph D and let D_1  D_2 denote the strong product of D_1 and D_2. In this paper, we obtain that the twin domination number of strong product of two directed cycles of length at least 2.Furthermore, we give a lower bound of the twin domination number of strong product of two digraphs, and prove that the twin domination number of strong product of the complete digraph and any digraph D equals the twin domination number of D.     

图的连通控制数与无赘数的一个不等式

设G是连通图,γ_C(G)和ir(G)分别表示G的连通控制数和无赘数。孙良于1990年证明了γ_c(G)≤4ir(G)—2,同时提出猜想γ_c(G)≤3ir(G)—2。本文进一步研究γ_c(G)与ir(G)的关系,并证得上述猜想成立。     

两类图的符号控制数

图G的符号控制数γs(G)有着许多重要的应用背景,因而确定其精确值有重要意义.Cm表示m个顶点的圈,n-Cm和n·Cm分别表示恰有一条公共边或一个公共顶点的n个Cm的拷贝.给出了n-Cm和n·Cm的符号控制数.     

关于图的减控制与符号控制

给定一个图G=(V,E),一个函数f:V→{-1,0,1}被称为G的减控制函数,如果对任意v∈V(G)均有∑_μ∈N[v]f(μ)≥1。G的减控制数定义为γ^-(G)=min{∑_v∈Vf(v)|f是G的减控制函数}。图G的符号控制函数的正如减控制函数,差别是广{-1,0,1}换成{-1,1}。符号控制数γ_s(G)是类似的。本文获得γ^-G)和γ_s(G)的一些下界。同时也证明并推广了 Jean Dunbar等提出的一个猜想,即对任意 n阶 2部图 G,均有γ^-(G)≥ 4(n+1^1/2-1)-n成立。     

图的局部减边控制数

引入局部减边控制函数和局部减边控制数的概念,得到了图的最小局部减边控制函数的性质,给出了局部减边控制数的最好上下界,确定了一些特殊图的局部减边控制数.最后得到了图的减边控制数的最好上界.     

Edge Domination in Graphs of Cubes

The signed edge domination number and the signed total edge domination number of a graph are considered; they are variants of the domination number and the total domination number. Some upper bounds for them are found in the case of the n-dimensional cube Q _n.     

关于图的符号边全控制数

引入了图的符号边全控制的概念,给出了一个连通图G的符号边全控制数γs′t(G)的下限,确定所有n阶树T的最小符号边全控制数,并刻划了满足γs′t(G)=E(G)的所有连通图G,最后还提出了一个关于γs′t(G)上界的猜想.     

On the doubly connected domination number of a graph

For a given connected graph G = (V, E), a set is a doubly connected dominating set if it is dominating and both 〈D〉 and 〈V (G)-D〉 are connected. The cardinality of the minimum doubly connected dominating set in G is the doubly connected domination number. We investigate several properties of doubly connected dominating sets and give some bounds on the doubly connected domination number.     

A New Bound on the Total Domination Subdivision Number

A set S of vertices of a graph G = (V, E) without isolated vertex is a total dominating set if every vertex of V(G) is adjacent to some vertex in S. The total domination number γ _t (G) is the minimum cardinality of a total dominating set of G. The total domination subdivision number is the minimum number of edges that must be subdivided (each edge in G can be subdivided at most once) in order to increase the total domination number. In this paper we prove that for every simple connected graph G of order n ≥ 3,

关于图的强符号全控制数

图的强符号全控制数有着许多重要的应用背景,因而确定其下界有重要的意义.本文提出了图的强符号全控制数的概念,在构造适当点集的基础上对其进行了研究,给出了：（1）一般图的强符号全控制数的5个独立可达的下界及达到其界值的图;（2）确定了圈、轮图、完全图、完全二部图的强符号全控制数的值.     

图的好符号星控制数

引入了图的好符号星控制的概念,求出了欧拉图、完全二部图、完全图和轮图的好符号星控制数,并改进了图的符号星控制数的两个上界.