Characterizations of trees with equal paired and double domination numbers

A paired-dominating set of a graph G is a dominating set of vertices whose induced subgraph has a perfect matching, and a double dominating set is a dominating set that dominates every vertex of G at least twice. We show that for trees, the paired-domination number is less than or equal to the double domination number, solving a conjecture of Chellali and Haynes. Then we characterize the trees having equal paired and double domination numbers.     

Locating and paired-dominating sets in graphs

In this paper, we continue the study of paired-domination in graphs introduced by Haynes and Slater [T.W. Haynes, P.J. Slater, Paired-domination in graphs, Networks 32 (1998), 199–206]. A paired-dominating set of a graph G with no isolated vertex is a dominating set S of vertices whose induced subgraph has a perfect matching. We consider paired-dominating sets which are also locating sets, that is distinct vertices of G are dominated by distinct subsets of the paired-dominating set. We consider three variations of sets which are paired-dominating and locating sets and investigate their properties.     

Partitioning the Vertices of a Graph into Two Total Dominating Sets

A total dominating set in a graph G is a set S of vertices of G such that every vertex in G is adjacent to a vertex of S. We study graphs whose vertex set can be partitioned into two total dominating sets. In particular, we develop several sufficient conditions for a graph to have a vertex partition into two total dominating sets. We also show that with the exception of the cycle on five vertices, every selfcomplementary graph with minimum degree at least two has such a partition.     

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.     

The diameter of paired-domination vertex critical graphs

In this paper we continue the study of paired-domination in graphs introduced by Haynes and Slater (Networks 32 (1998), 199–206). A paired-dominating set of a graph G with no isolated vertex is a dominating set of vertices whose induced subgraph has a perfect matching. The paired-domination number of G, denoted by γ _pr(G), is the minimum cardinality of a paired-dominating set of G. The graph G is paired-domination vertex critical if for every vertex v of G that is not adjacent to a vertex of degree one, γ _pr(G − v) < γ _pr(G). We characterize the connected graphs with minimum degree one that are paired-domination vertex critical and we obtain sharp bounds on their maximum diameter. We provide an example which shows that the maximum diameter of a paired-domination vertex critical graph is at least 3/2 (γ _pr(G) − 2). For γ _pr(G) ⩽ 8, we show that this lower bound is precisely the maximum diameter of a paired-domination vertex critical graph. The first author was supported in part by the South African National Research Foundation and the University of KwaZulu-Natal, the second author was supported by the Natural Sciences and Engineering Research Council of Canada.     

A Characterization of Cubic Graphs with Paired-Domination Number Three-Fifths Their Order

A paired-dominating set of a graph is a dominating set of vertices whose induced subgraph has a perfect matching, while the paired-domination number is the minimum cardinality of a paired-dominating set in the graph. Recently, Chen et al. (Acta Math Sci Ser A Chin Ed 27(1):166–170, 2007) proved that a cubic graph has paired-domination number at most three-fifths the number of vertices in the graph. In this paper, we show that the Petersen graph is the only connected cubic graph with paired-domination number three-fifths its order.     

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.     

A polynomial-time algorithm for the paired-domination problem on permutation graphs

A set S of vertices in a graph H=(V,E) with no isolated vertices is a paired-dominating set of H if every vertex of H is adjacent to at least one vertex in S and if the subgraph induced by S contains a perfect matching. Let G be a permutation graph and π be its corresponding permutation. In this paper we present an O(mn) time algorithm for finding a minimum cardinality paired-dominating set for a permutation graph G with n vertices and m edges.     

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     

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.     

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.     

Hereditary dominating pair graphs

An asteroidal triple (AT) is a set of vertices such that each pair of vertices is joined by a path that avoids the neighborhood of the third. Every AT-free graph contains a dominating pair, a pair of vertices such that for every path between them, every vertex of the graph is within distance one of the path. We say that a graph is a hereditary dominating pair (HDP) graph if each of its connected induced subgraphs contains a dominating pair. In this paper we introduce the notion of frame HDP graphs in order to capture the structure of HDP graphs that contain asteroidal triples. We also determine the maximum diameter of frame HDP graphs.     

Augmenting a graph of minimum degree 2 to have two disjoint total dominating sets

A total dominating set of a graph is a set of vertices such that every vertex is adjacent to a vertex in the set. We show that given a graph of order n with minimum degree at least 2, one can add at most edges such that the resulting graph has two disjoint total dominating sets, and this bound is best possible.     

Vertices contained in all or in no minimum total dominating set of a tree

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. We characterize the set of vertices of a tree that are contained in all, or in no, minimum total dominating sets of the tree.     

A note on power domination in grid graphs

The problem of monitoring an electric power system by placing as few measurement devices in the system as possible is closely related to the well known vertex covering and dominating set problems in graphs (see [T.W. Haynes, S.M. Hedetniemi, S.T. Hedetniemi, M.A. Henning, Power domination in graphs applied to electrical power networks, SIAM J. Discrete Math. 15(4) (2002) 519-529]). A set S of vertices is defined to be a power dominating set of a graph if every vertex and every edge in the system is monitored by the set S (following a set of rules for power system monitoring). The minimum cardinality of a power dominating set of a graph is its power domination number. In this paper, we determine the power domination number of an n×m grid graph.     

On matching and total domination in graphs

A set M of edges of a graph G is a matching if no two edges in M are incident to the same vertex. A set S of vertices in G is a total dominating set of G if every vertex of G is adjacent to some vertex in S. The matching number is the maximum cardinality of a matching of G, while the total domination number of G is the minimum cardinality of a total dominating set of G. In this paper, we investigate the relationships between the matching and total domination number of a graph. We observe that the total domination number of every claw-free graph with minimum degree at least three is bounded above by its matching number, and we show that every k-regular graph with k?3 has total domination number at most its matching number. In general, we show that no minimum degree is sufficient to guarantee that the matching number and total domination number are comparable.     

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.     

无向路图和块图上的混合控制

图G=(V,E)的一个混合控制集是一个满足如下条件的集合DV∪E:不在D中的每个点或每条边都相邻或关联于D中的至少一个点或一条边.确定图的最小基数的混合控制集的问题称为混合控制问题.本文研究混合控制问题的算法复杂性,证明了混合控制问题在无向路图上是NP-完全的,但在块图上有线性时间算法.无向路图和块图都是弦图的子类,又是树的母类.     

Paired-Domination in Subdivided Star-Free Graphs

A set S of vertices in a graph G is a paired-dominating set of G if every vertex of G is adjacent to some vertex in S and if the subgraph induced by S contains a perfect matching. The paired-domination number of G, denoted by γ _pr(G), is the minimum cardinality of a paired-dominating set of G. In [Dorbec P, Gravier S, Henning MA, J Comb Optim 14(1):1–7, 2007], the authors gave tight bounds for paired-dominating sets of generalized claw-free graphs. Yet, the critical cases are not claws but subdivided stars. We here give a bound for graphs containing no induced subdivided stars, depending on the size of the star.     

Paired-Domination in <Emphasis Type="Italic">P</Emphasis><Subscript>5</Subscript>-Free Graphs

A set S of vertices in a graph G is a paired-dominating set of G if every vertex of G is adjacent to some vertex in S and if the subgraph induced by S contains a perfect matching. The paired-domination number of G, denoted by , is the minimum cardinality of a paired-dominating set of G. In [1], the authors gave tight bounds for paired-dominating sets of generalized claw-free graphs. Yet, the critical cases are not claws but subdivided stars. We here give a bound for graphs containing no induced P ₅, which seems to be the critical case.