On connected domination in unit ball graphs

Given a simple undirected graph, the minimum connected dominating set problem is to find a minimum cardinality subset of vertices D inducing a connected subgraph such that each vertex outside D has at least one neighbor in D. Approximations of minimum connected dominating sets are often used to represent a virtual routing backbone in wireless networks. This paper first proposes a constant-ratio approximation algorithm for the minimum connected dominating set problem in unit ball graphs and then introduces and studies the edge-weighted bottleneck connected dominating set problem, which seeks a minimum edge weight in the graph such that the corresponding bottleneck subgraph has a connected dominating set of size k. In wireless network applications this problem can be used to determine an optimal transmission range for a network with a predefined size of the virtual backbone. We show that the problem is hard to approximate within a factor better than 2 in graphs whose edge weights satisfy the triangle inequality and provide a 3-approximation algorithm for such graphs. We also show that for fixed k the problem is polynomially solvable in unit disk and unit ball graphs.     

A PTAS for minimum <Emphasis Type="Italic">d</Emphasis>-hop connected dominating set in growth-bounded graphs

In this paper, we design the first polynomial time approximation scheme for d-hop connected dominating set (d-CDS) problem in growth-bounded graphs, which is a general type of graphs including unit disk graph, unit ball graph, etc. Such graphs can represent majority types of existing wireless networks. Our algorithm does not need geometric representation (e.g., specifying the positions of each node in the plane) beforehand. The main strategy is clustering partition. We select the d-CDS for each subset separately, union them together, and then connect the induced graph of this set. We also provide detailed performance and complexity analysis.     

PTAS for the minimum weighted dominating set in growth bounded graphs

The minimum weighted dominating set (MWDS) problem is one of the classic NP-hard optimization problems in graph theory with applications in many fields such as wireless communication networks. MWDS in general graphs has been showed not to have polynomial-time constant-approximation if ${\mathcal{NP} \neq \mathcal{P}}$ . Recently, several polynomial-time constant-approximation SCHEMES have been designed for MWDS in unit disk graphs. In this paper, using the local neighborhood-based scheme technique, we present a PTAS for MWDS in polynomial growth bounded graphs with bounded degree constraint.     

Least domination in a graph

The least domination number γ_L of a graph G is the minimum cardinality of a dominating set of G whose domination number is minimum. The least point covering number _L of G is the minimum cardinality of a total point cover (point cover including every isolated vertex of G) whose total point covering number is minimum. We prove a conjecture of Sampathkumar saying that in every connected graph of order n 2. We disprove another one saying that γ_{L L} in every graph but instead of it, we establish the best possible inequality . Finally, in relation with the minimum cardinality γ_t of a dominating set without isolated vertices (total dominating set), we prove that the ratio γ_L/γ_t can be in general arbitrarily large, but remains bounded by if we restrict ourselves to the class of trees.     

PTAS for the minimum k-path connected vertex cover problem in unit disk graphs

In the Minimum k-Path Connected Vertex Cover Problem (MkPCVCP), we are given a connected graph G and an integer k ≥ 2, and are required to find a subset C of vertices with minimum cardinality such that each path with length k ? 1 has a vertex in C, and moreover, the induced subgraph G[C] is connected. MkPCVCP is a generalization of the minimum connected vertex cover problem and has applications in many areas such as security communications in wireless sensor networks. MkPCVCP is proved to be NP-complete. In this paper, we give the first polynomial time approximation scheme (PTAS) for MkPCVCP in unit disk graphs, for every fixed k ≥ 2.     

Minimum Clique Partition in Unit Disk Graphs

The minimum clique partition (MCP) problem is that of partitioning the vertex set of a given graph into a minimum number of cliques. Given n points in the plane, the corresponding unit disk graph (UDG) has these points as vertices, and edges connecting points at distance at most 1. MCP in UDGs is known to be NP-hard and several constant factor approximations are known, including a recent PTAS. We present two improved approximation algorithms for MCP in UDGs with a realization: (I) A polynomial time approximation scheme (PTAS) running in time n^O(1/e2){n^{O(1/\varepsilon^2)}}. This improves on a previous PTAS with n^O(1/e4){n^{O(1/\varepsilon^4)}} running time by Pirwani and Salavatipour (arXiv:0904.2203v1, 2009). (II) A randomized quadratic-time algorithm with approximation ratio 2.16. This improves on a ratio 3 algorithm with O(n ²) running time by Cerioli et al. (Electron. Notes Discret. Math. 18:73–79, 2004).     

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.     

An extremal problem for edge domination insensitive graphs

A connected graph is edge domination insensitive if the domination number is unchanged when any single edge is removed. The minimum number of edges required by such a graph is determined. Similar results are given when the graph must remain connected upon any edge's removal and when the dominating set must remain fixed.     

Bounds related to domination in graphs with minimum degree two

A dominating set for a graph G = (V,E) is a subset of vertices V′ ⊆ V such that for all v E V − V′ there exists some u E V′ for which {v, u} E E. The domination number of G is the size of its smallest dominating set(s). We show that for almost all connected graphs with minimum degree at least 2 and q edges, the domination number is bounded by (q + 1)/3. From this we derive exact lower bounds for the number of edges of a connected graph with minimum degree at least 2 and a given domination number. We also generalize the bound to k-restricted domination numbers; these measure how many vertices are necessary to dominate a graph if an arbitrary set of k vertices must be incluced in the dominating set. © 1997 John Wiley & Sons, Inc. J Graph Theory 25: 139–152, 1997     

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.     

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.     

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.     

Vertex-edge domination in graphs

Aequationes mathematicae - We establish that for any connected graph G of order $$n \ge 6$$ , a minimum vertex-edge dominating set of G has at most n/3 vertices, thus affirmatively answering the...     

Power domination in 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 domination problem in graphs. In 1998, Haynes et al. considered the graph theoretical representation of this problem as a variation of the domination problem. They defined a set S 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 power domination number γ_P(G) of a graph G is the minimum cardinality of a power dominating set of G. In this paper, we present upper bounds on the power domination number for a connected graph with at least three vertices and a connected claw-free cubic graph in terms of their order. The extremal graphs attaining the upper bounds are also characterized.     

Connected Domatic Number in Planar Graphs

A dominating set in a graph G is a connected dominating set of G if it induces a connected subgraph of G. The connected domatic number of G is the maximum number of pairwise disjoint, connected dominating sets in V(G). We establish a sharp lower bound on the number of edges in a connected graph with a given order and given connected domatic number. We also show that a planar graph has connected domatic number at most 4 and give a characterization of planar graphs having connected domatic number 3.     

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.     

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.     

Bounds on Laplacian eigenvalues related to total and signed domination of graphs

A total dominating set in a graph G is a subset X of V (G) such that each vertex of V (G) is adjacent to at least one vertex of X. The total domination number of G is the minimum cardinality of a total dominating set. A function f: V (G) → {−1, 1} is a signed dominating function (SDF) if the sum of its function values over any closed neighborhood is at least one. The weight of an SDF is the sum of its function values over all vertices. The signed domination number of G is the minimum weight of an SDF on G. In this paper we present several upper bounds on the algebraic connectivity of a connected graph in terms of the total domination and signed domination numbers of the graph. Also, we give lower bounds on the Laplacian spectral radius of a connected graph in terms of the signed domination number of the graph.     

An exact algorithm for minimum CDS with shortest path constraint in wireless networks

In this paper, we study a minimum connected dominating set problem (CDS) in wireless networks, which selects a minimum CDS with property that all intermediate nodes inside every pairwise shortest path should be included. Such a minimum CDS (we name this problem as SPCDS) is an important tache of some other algorithms for constructing a minimum CDS. We prove that finding such a minimum SPCDS can be achieved in polynomial time and design an exact algorithm with time complexity O(δ ² n), where δ is the maximum node degree in communication graph.     

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.