极大全控点临界图

图G的点集S如果满足:VG-S(或VG)中每个点相邻于S中的某个点(或而不是它本身),则称点集S是一个控制集(或全控制集).图G的所有控制集(或全控制集)中最小基数的控制集(或全控制集)中的点数,称为控制数(或全控数),记为γ(G)(或γt(G)).在这篇文章中我们特征化γt-临界图且满足γt(G)=n-Δ(G)的图特征,这回答了Goddard等人提出的一个问题.     

Total restrained bondage in graphs

A subset S of vertices of a graph G with no isolated vertex is a total restrained dominating set if every vertex is adjacent to a vertex in S and every vertex in V (G) S is also adjacent to a vertex in V (G) S. The total restrained domination number of G is the minimum cardinality of a total restrained dominating set of G. In this paper we initiate the study of total restrained bondage in graphs. The total restrained bondage number in a graph G with no isolated vertex, is the minimum cardinality of a subset of edges E such that G E has no isolated vertex and the total restrained domination number of G E is greater than the total restrained domination number of G. We obtain several properties, exact values and bounds for the total restrained bondage number of a graph.     

无爪3 -正则图的独立数

如果图G的一个集合X中任两个点不相邻, 则称 X 为独立集合. 如果 N[X]=V(G), 则称X是一个控制集合. i(G)(β(G))分别表示所有极大独立集合的最小(最大)基数. γ(G)(Γ(G))表示所有极小控制集合的最小(最大)基数. 在这篇论文中, 作者证明如下结论: (1) 如果 G ∈R 且G 是n阶3 -正则图, 则 γ(G)= i(G), β(G)=n/3. (2) 每个n阶连通无爪3 -正则图 G, 如果 G(G≠ K₄) 且不含诱导子图K₄-e, 则 β(G) =n/3.     

本刊英文版Vol.29(2013),No.6论文摘要

<正>Total Restrained Bondage in Graphs Nader JAFARI RAD Roslan HASNI Joanna RACZEK Lutz VOLKMANN Abstract A subset S of vertices of a graph G with no isolated vertex is a total restrained dominating set if every vertex is adjacent to a vertex in S and every vertex in V(G)-S is also adjacent to a vertex in V(G)-S.The total restrained domination number of G is the minimum cardinality of a total restrained dominating set of G.In this paper we initiate the study of     

有关循环图C(n;{1,k})的独立数的一些结果

令G=(V(G),E(G))是一个简单有限无向图.如果V(G)的子集S中任意两个顶点均不相邻,则S是图G的一个独立集.顶点独立集大小的最大值,称为图G的独立数,记作α(G).本文研究了循环图C(n;{1,k})的独立数问题,并给出了当k=2,3,4,5时的准确值.     

Disjoint independent dominating sets in graphs

A set X of vertices of G is an independent dominating set if no two vertices of X are adjacent and each vertex not in X is adjacent to at least one vertex in X. Independent dominating sets of G are cliques of the complement $\text{G}$ of G and conversely.This work is concerned with the existence of disjoint independent dominating sets in a graph G. A new parameter, the maximum number of disjoint independent dominating sets in G, is studied and the class of graphs whose vertex sets partition into independent dominating sets is investigated.     

阶数极大的临界全控制图

称一个没有孤立点的图G 为临界全控制图, 如果G 满足对于任何一个不与悬挂点相邻的顶点v, G - v 的全控制数都小于G 的全控制数. 如果G 的全控制数记为γ_t, 则称这样的临界全控制图G 为γ_t- 临界的. 如果G 是γ_t- 临界的, 且阶数为n, 则n ≤ Δ(G)(γ_t(G)- 1) + 1, 其中Δ(G) 是G 的最大度. 本文将证明对γ_t = 3, 这个阶数的上界是紧的, 并给出所有满足n = Δ(G)(γ_t(G)- 1) + 1 的3-γ_t- 临界图.     

Total domination dot-stable graphs

A set S of vertices in a graph G is a total dominating set if every vertex of G is adjacent to some vertex in S. The minimum cardinality of a total dominating set of G is the total domination number of G. Two vertices of G are said to be dotted (identified) if they are combined to form one vertex whose open neighborhood is the union of their neighborhoods minus themselves. We note that dotting any pair of vertices cannot increase the total domination number. Further we show it can decrease the total domination number by at most 2. A graph is total domination dot-stable if dotting any pair of adjacent vertices leaves the total domination number unchanged. We characterize the total domination dot-stable graphs and give a sharp upper bound on their total domination number. We also characterize the graphs attaining this bound.     

The Rainbow Vertex-disconnection in Graphs

Let G be a nontrivial connected and vertex-colored graph. A subset X of the vertex set of G is called rainbow if any two vertices in X have distinct colors. The graph G is called rainbow vertex-disconnected if for any two vertices x and y of G, there exists a vertex subset S of G such that when x and y are nonadjacent, S is rainbow and x and y belong to different components of G-S; whereas when x and y are adjacent, S + x or S + y is rainbow and x and y belong to different components of(G-xy)-S. For a connected graph G, the rainbow vertex-disconnection number of G, denoted by rvd(G), is the minimum number of colors that are needed to make G rainbow vertexdisconnected. In this paper, we characterize all graphs of order n with rainbow vertex-disconnection number k for k ∈ {1, 2, n}, and determine the rainbow vertex-disconnection numbers of some special graphs. Moreover, we study the extremal problems on the number of edges of a connected graph G with order n and rvd(G) = k for given integers k and n with 1 ≤ k ≤ n.     

Domination-balanced graphs

A set D of vertices in a graph is said to be a dominating set if every vertex not in D is adjacent to some vertex in D. The domination number β(G) of a graph G is the size of a smallest dominating set. G is called domination balanced if its vertex set can be partitioned into β(G) subsets so that each subset is a smallest dominating set of the complement G of G. The purpose of this paper is to characterize these graphs.     

图的半强积的邻点可区别染色

两个简单图G与H的半强积G·H是具有顶点集V(G)×V(H)的简单图,其中两个顶点(u,v)与(u',v')相邻当且仅当u=u'且vv'∈E(H),或uu'∈E(G)且vv'∈E(H).图的邻点可区别边(全)染色是指相邻点具有不同色集的正常边(全)染色.统称图的邻点可区别边染色与邻点可区别全染色为图的邻点可区别染色.图G的邻点可区别染色所需的最少的颜色数称为邻点可区别染色数,并记为X_a~((r))(G),其中r=1,2,且X_a~((1))(G)与X_a~((2))(G)分别表示G的邻点可区别的边色数与全色数.给出了两个简单图的半强积的邻点可区别染色数的一个上界,并证明了该上界是可达的.然后,讨论了两个树的不同半强积具有相同邻点可区别染色数的充分必要条件.另外,确定了一类图与完全图的半强积的邻点可区别染色数的精确值.     

图的双控制的一些新结果

图G=(V,E)的每个顶点控制它的闭邻域的每个顶点.S是一个顶点子集合,如果G的每一个顶点至少被S中的两个顶点控制,则称S是G的一个双控制集.把双控制集的最小基数称为双控制数,记为dd(G).本文探讨了双控制数和其它控制参数的一些新关系,推广了[1]的一些结果.并且给出了双控制数的Nordhaus-Gaddum类型的结果.     

Total domination stable graphs upon edge addition

A set S of vertices in a graph G is a total dominating set if every vertex of G is adjacent to some vertex in S. The minimum cardinality of a total dominating set of G is the total domination number of G. A graph is total domination edge addition stable if the addition of an arbitrary edge has no effect on the total domination number. In this paper, we characterize total domination edge addition stable graphs. We determine a sharp upper bound on the total domination number of total domination edge addition stable graphs, and we determine which combinations of order and total domination number are attainable. We finish this work with an investigation of claw-free total domination edge addition stable graphs.     

On the Independent Domination Number of Regular Graphs

A set S of vertices in a graph G is an independent dominating set of G if S is an independent set and every vertex not in S is adjacent to a vertex in S. In this paper, we consider questions about independent domination in regular graphs.     

Total domination changing and stable graphs upon vertex removal

A set S of vertices in a graph G is a total dominating set if every vertex of G is adjacent to some vertex in S. The minimum cardinality of a total dominating set of G is the total domination number of G. A graph is total domination vertex removal stable if the removal of an arbitrary vertex leaves the total domination number unchanged. On the other hand, a graph is total domination vertex removal changing if the removal of an arbitrary vertex changes the total domination number. In this paper, we study total domination vertex removal changing and stable graphs.     

Total domination critical and stable graphs upon edge removal

A set S of vertices in a graph G is a total dominating set of G if every vertex of G is adjacent to some vertex in S. The minimum cardinality of a total dominating set of G is the total domination number of G. A graph is total domination edge critical if the removal of any arbitrary edge increases the total domination number. On the other hand, a graph is total domination edge stable if the removal of any arbitrary edge has no effect on the total domination number. In this paper, we characterize total domination edge critical graphs. We also investigate various properties of total domination edge stable graphs.     

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.     

Domination versus disjunctive domination in graphs

A dominating set in a graph G is a set S of vertices of G such that every vertex not in S is adjacent to a vertex of S. The domination number of G is the minimum cardinality of a dominating set of G. For a positive integer b, a set S of vertices in a graph G is a b-disjunctive dominating set in G if every vertex v not in S is adjacent to a vertex of S or has at least b vertices in S at distance 2 from it in G. The b-disjunctive domination number of G is the minimum cardinality of a b-disjunctive dominating set. In this paper, we continue the study of disjunctive domination in graphs. We present properties of b-disjunctive dominating sets in a graph. A characterization of minimal b-disjunctive dominating sets is given. We obtain bounds on the ratio of the domination number and the b-disjunctive domination number for various families of graphs, including regular graphs and trees.     

Fibonacci Polynomials and Parity Domination in Grid Graphs

 A non-empty set of vertices is called an even dominating set if each vertex in the graph is adjacent to an even number of vertices in the set (adjacency is reflexive). In this paper, the Fibonacci polynomials are studied over GF(2) with particular emphasis on their divisibility properties and their relation to the existence of even dominating sets in grid graphs and properties of a corresponding recurrence. Received: March 15, 1999 Final version received: November 8, 1999     

Total domination in 2‐connected graphs and in graphs with no induced 6‐cycles

A set S of vertices in a graph G is a total dominating set of G if every vertex of G is adjacent to some vertex in S. The minimum cardinality of a total dominating set of G is the total domination number γ_t(G) of G. It is known [J Graph Theory 35 (2000), 21–45] that if G is a connected graph of order n > 10 with minimum degree at least 2, then γ_t(G) ≤ 4n/7 and the (infinite family of) graphs of large order that achieve equality in this bound are characterized. In this article, we improve this upper bound of 4n/7 for 2‐connected graphs, as well as for connected graphs with no induced 6‐cycle. We prove that if G is a 2‐connected graph of order n > 18, then γ_t(G) ≤ 6n/11. Our proof is an interplay between graph theory and transversals in hypergraphs. We also prove that if G is a connected graph of order n > 18 with minimum degree at least 2 and no induced 6‐cycle, then γ_t(G) ≤ 6n/11. Both bounds are shown to be sharp. © 2008 Wiley Periodicals, Inc. J Graph Theory 60: 55–79, 2009