Dominating and total dominating partitions in cubic graphs

In this paper, we continue the study of domination and total domination in cubic graphs. It is known [Henning M.A., Southey J., A note on graphs with disjoint dominating and total dominating sets, Ars Combin., 2008, 89, 159–162] that every cubic graph has a dominating set and a total dominating set which are disjoint. In this paper we show that every connected cubic graph on nvertices has a total dominating set whose complement contains a dominating set such that the cardinality of the total dominating set is at most (n+2)/2, and this bound is essentially best possible.     

Efficient open domination in Cayley graphs

Efficient open dominating sets in bipartite Cayley graphs are characterized in terms of covering projections. Necessary and sufficient conditions for the existence of efficient open dominating sets in certain circulant Harary graphs are given. Chains of efficient dominating sets, and of efficient open dominating sets, in families of circulant graphs are described as an application.     

Graphs with disjoint dominating and paired-dominating sets

A dominating set of a graph is a set of vertices such that every vertex not in the set is adjacent to a vertex in the set, while a paired-dominating set of a graph is a dominating set such that the subgraph induced by the dominating set contains a perfect matching. In this paper, we show that no minimum degree is sufficient to guarantee the existence of a disjoint dominating set and a paired-dominating set. However, we prove that the vertex set of every cubic graph can be partitioned into a dominating set and a paired-dominating set.     

Efficient edge domination in regular graphs

An induced matching of a graph G is a matching having no two edges joined by an edge. An efficient edge dominating set of G is an induced matching M such that every other edge of G is adjacent to some edge in M. We relate maximum induced matchings and efficient edge dominating sets, showing that efficient edge dominating sets are maximum induced matchings, and that maximum induced matchings on regular graphs with efficient edge dominating sets are efficient edge dominating sets. A necessary condition for the existence of efficient edge dominating sets in terms of spectra of graphs is established. We also prove that, for arbitrary fixed p≥3, deciding on the existence of efficient edge dominating sets on p-regular graphs is NP-complete.     

A note about the dominating circuit conjecture

The dominating circuit conjecture states that every cyclically 4-edge-connected cubic graph has a dominating circuit. We show that this is equivalent to the statement that any two edges of such a cyclically 4-edge-connected graph are contained in a dominating circuit.     

Wireless networking,dominating and packing

Recently, there are many papers published in the literature on study of dominating sets, connected dominating sets and their variations motivated from various applications in wireless networks and social networks. In this article, we survey those developments for wireless networking, dominating, and packing problems.     

On parallelizing a greedy heuristic for finding small dominant sets

We analyse a greedy heuristic for finding small dominating sets in graphs: bounds on the size of the dominating set so produced had previously been derived in terms of the size of a smallest dominating set and the number of vertices and edges in the graph, respectively, We show that computing the resulting small dominating set isP-hard and so cannot be done efficiently in parallel (in the context of the PRAM model of parallel computation). We also consider a related non-deterministic greedy heuristic.     

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.     

Dominating Sets for Convex Functions with Some Applications

A number of optimization methods require as a first step the construction of a dominating set (a set containing an optimal solution) enjoying properties such as compactness or convexity. In this paper, we address the problem of constructing dominating sets for problems whose objective is a componentwise nondecreasing function of (possibly an infinite number of) convex functions, and we show how to obtain a convex dominating set in terms of dominating sets of simpler problems. The applicability of the results obtained is illustrated with the statement of new localization results in the fields of linear regression and location.     

图的最小控制树问题（英文）

A dominating tree T of a graph G is a subtree of G which contains at least one neighbor of each vertex of G.The minimum dominating tree problem is to find a dominating tree of G with minimum number of vertices,which is an NP-hard problem.This paper studies some polynomially solvable cases,including interval graphs,Halin graphs,special outer-planar graphs and others.     

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.     

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.     

Disjoint dominating and total dominating sets in graphs

It has been shown [M.A. Henning, J. Southey, A note on graphs with disjoint dominating and total dominating sets, Ars Combin. 89 (2008) 159-162] that every connected graph with minimum degree at least two that is not a cycle on five vertices has a dominating set D and a total dominating set T which are disjoint. We characterize such graphs for which D∪T necessarily contains all vertices of the graph and that have no induced cycle on five vertices.     

A linear algorithm for the domination number of a series-parallel graph

A vertex u in an undirected graph G = (V, E) is said to dominate all its adjacent vertices and itself. A subset D of V is a dominating set in G if every vertex in G is dominated by a vertex in D, and is a minimum dominating set in G if no other dominating set in G has fewer vertices than D. The domination number of G is the cardinality of a minimum dominating set in G.The problem of determining, for a given positive integer k and an undirected graph G, whether G has a dominating set D in G satisfying ¦D¦ ≤ k, is a well-known NP-complete problem. Cockayne have presented a linear time algorithm for finding a minimum dominating set in a tree. In this paper, we will present a linear time algorithm for finding a minimum dominating set in a series-parallel graph.     

Dominating sets in social network graphs

This paper presents an approach to the study of the structure of social networks using dominating sets of graphs. A method is presented for partitioning the vertices of a graph using dominating vertices. For certain classes of graphs this is helpful in determining the underlying structure of the corresponding social network. An extension of this technique provides a method of studying the structure of directed graphs and directed social networks. Minimum dominating sets are related to statuses and structurally equivalent sets.     

Connected domination of regular graphs

A dominating setD of a graph G is a subset of V(G) such that for every vertex v∈V(G), either v∈D or there exists a vertex u∈D that is adjacent to v in G. Dominating sets of small cardinality are of interest. A connected dominating setC of a graph G is a dominating set of G such that the subgraph induced by the vertices of C in G is connected. A weakly-connected dominating setW of a graph G is a dominating set of G such that the subgraph consisting of V(G) and all edges incident with vertices in W is connected. In this paper we present several algorithms for finding small connected dominating sets and small weakly-connected dominating sets of regular graphs. We analyse the average-case performance of these heuristics on random regular graphs using differential equations, thus giving upper bounds on the size of a smallest connected dominating set and the size of a smallest weakly-connected dominating set of random regular graphs.     

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.     

Line domination in graphs

A setS of lines is a line dominating set if every line not inS is adjacent to some line ofS. The line domination number of a graph is the cardinality of a minimum line dominating set. In this paper we study the line dominating sets and obtain bounds for the line domination number. Also, Nordhaus-Gaddum type results are obtained for the line domination number and the line domatic number.     

Liar’s domination in graphs

Assume that each vertex of a graph G is the possible location for an “intruder” such as a thief, or a saboteur, a fire in a facility or some possible processor fault in a computer network. A device at a vertex v is assumed to be able to detect the intruder at any vertex in its closed neighborhood N[v]and to identify at which vertex inN[v] the intruder is located. One must then have a dominating set S⊆V(G), a set with ∪_v∈SN[v]=V(G), to be able to identify any intruder’s location. If any one device can fail to detect the intruder, then one needs a double-dominating set. This paper considers liar’s dominating sets, sets that can identify an intruder’s location even when any one device in the neighborhood of the intruder vertex can lie, that is, any one device in the neighborhood of the intruder vertex can misidentify any vertex in its closed neighborhood as the intruder location. Liar’s dominating sets lie between double dominating sets and triple dominating sets because every triple dominating set is a liar’s dominating set, and every liar’s dominating set must double dominate.     

A note on the two-stage hybrid flow shop problem with dedicated machines

In this paper we introduce a new dominance rule for the two-stage hybrid flow shop problem with dedicated machines. The rule is then used to construct a dominating set. The efficiency of the proposed rule is shown through an analysis of the dominating set cardinality.