Trees with Equal Domination and Restrained Domination Numbers

Let G = (V,E) be a graph and let S V. The set S is a packing in G if the vertices of S are pairwise at distance at least three apart in G. The set S is a dominating set (DS) if every vertex in V − S is adjacent to a vertex in S. Further, if every vertex in V − S is also adjacent to a vertex in V − S, then S is a restrained dominating set (RDS). The domination number of G, denoted by γ(G), is the minimum cardinality of a DS of G, while the restrained domination number of G, denoted by γ_r(G), is the minimum cardinality of a RDS of G. The graph G is γ-excellent if every vertex of G belongs to some minimum DS of G. A constructive characterization of trees with equal domination and restrained domination numbers is presented. As a consequence of this characterization we show that the following statements are equivalent: (i) T is a tree with γ(T)=γ_r(T); (ii) T is a γ-excellent tree and T ≠ K₂; and (iii) T is a tree that has a unique maximum packing and this set is a dominating set of T. We show that if T is a tree of order n with ℓ leaves, then γ_r(T) ≤ (n + ℓ + 1)/2, and we characterize those trees achieving equality.     

Proof of a conjecture on game domination

Let G(V, E) be a simple, undirected graph where V is the set of vertices and E is the set of edges. A b‐dimensional cube is a Cartesian product I₁×I₂×···×I_b, where each I_i is a closed interval of unit length on the real line. The cubicity of G, denoted by cub(G), is the minimum positive integer b such that the vertices in G can be mapped to axis parallel b‐dimensional cubes in such a way that two vertices are adjacent in G if and only if their assigned cubes intersect. An interval graph is a graph that can be represented as the intersection of intervals on the real line—i.e. the vertices of an interval graph can be mapped to intervals on the real line such that two vertices are adjacent if and only if their corresponding intervals overlap. Suppose S(m) denotes a star graph on m+1 nodes. We define claw number ψ(G) of the graph to be the largest positive integer m such that S(m) is an induced subgraph of G. It can be easily shown that the cubicity of any graph is at least ?log₂ψ(G)?. In this article, we show that for an interval graph G ?log₂ψ(G)??cub(G)??log₂ψ(G)?+2. It is not clear whether the upper bound of ?log₂ψ(G)?+2 is tight: till now we are unable to find any interval graph with cub(G)>?log₂ψ(G)?. We also show that for an interval graph G, cub(G)??log₂α?, where α is the independence number of G. Therefore, in the special case of ψ(G)=α, cub(G) is exactly ?log₂α₂?. The concept of cubicity can be generalized by considering boxes instead of cubes. A b‐dimensional box is a Cartesian product I₁×I₂×···×I_b, where each I_i is a closed interval on the real line. The boxicity of a graph, denoted box(G), is the minimum k such that G is the intersection graph of k‐dimensional boxes. It is clear that box(G)?cub(G). From the above result, it follows that for any graph G, cub(G)?box(G)?log₂α?. © 2010 Wiley Periodicals, Inc. J Graph Theory 65: 323–333, 2010     

Restricted power domination and fault-tolerant power domination on grids

The power domination problem is to find a minimum placement of phase measurement units (PMUs) for observing the whole electric power system, which is closely related to the classical domination problem in graphs. For a graph G=(V,E), the power domination number of G is the minimum cardinality of a set S⊆V such that PMUs placed on every vertex of S results in all of V being observed. A vertex with a PMU observes itself and all its neighbors, and if an observed vertex with degree d>1 has only one unobserved neighbor, then the unobserved neighbor becomes observed. Although the power domination problem has been proved to be NP-complete even when restricted to some special classes of graphs, Dorfling and Henning in [M. Dorfling, M.A. Henning, A note on power domination in grid graphs, Discrete Applied Mathematics 154 (2006) 1023-1027] showed that it is easy to determine the power domination number of an n×m grid. Their proof provides an algorithm for giving a minimum placement of PMUs. In this paper, we consider the situation in which PMUs may only be placed within a restricted subset of V. Then, we present algorithms to solve this restricted type of power domination on grids under the conditions that consecutive rows or columns form a forbidden zone. Moreover, we also deal with the fault-tolerant measurement placement in the designed scheme and provide approximation algorithms when the number of faulty PMUs does not exceed 3.     

A survey of selected recent results on total domination in graphs

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. In this paper, we offer a survey of selected recent results on total domination in graphs.     

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．     

Total Restrained Domination in Cubic Graphs

A set S of vertices in a graph G = (V, E) is a total restrained dominating set (TRDS) of G if every vertex of G is adjacent to a vertex in S and every vertex of V − S is adjacent to a vertex in V − S. The total restrained domination number of G, denoted by γ _tr (G), is the minimum cardinality of a TRDS of G. Let G be a cubic graph of order n. In this paper we establish an upper bound on γ _tr (G). If adding the restriction that G is claw-free, then we show that γ _tr (G) = γ _t (G) where γ _t (G) is the total domination number of G, and thus some results on total domination in claw-free cubic graphs are valid for total restrained domination. Research was partially supported by the NNSF of China (Nos. 60773078, 10832006), the ShuGuang Plan of Shanghai Education Development Foundation (No. 06SG42) and Shanghai Leading Academic Discipline Project (No. S30104).     

On edge domination numbers of graphs

Let and be the signed edge domination number and signed star domination number of G, respectively. We prove that holds for all graphs G without isolated vertices, where n=|V(G)|?4 and m=|E(G)|, and pose some problems and conjectures.     

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,

Constructive characterizations of (γp,γ)- and (γp,γpr)-trees

Let G = (V,E) be a graph without isolated vertices.A set S V is a domination set of G if every vertex in V - S is adjacent to a vertex in S,that is N[S] = V.The domination number of G,denoted by γ(G),is the minimum cardinality of a domination set of G.A set S C V is a paired-domination set of G if S is a domination set of G and the induced subgraph G[S] has a perfect matching.The paired-domination number,denoted by γpr(G),is defined to be the minimum cardinality of a paired-domination set S in G.A subset S V is a power domination set of G if all vertices of V can be observed recursively by the following rules: (i) all vertices in N[S] are observed initially,and (ii) if an observed vertex u has all neighbors observed except one neighbor v,then v is observed (by u).The power domination number,denoted by γp(G),is the minimum cardinality of a power domination set of G.In this paper,the constructive characterizations for trees with γp = γ and γpr = γp are provided respectively.     

关于树T的γ/γc的界

控制γ和连通控制数γc是图的两个重要的控制参数，本文通过对树中的点进行恰当分类，给出了树中的γ/γc值的最好界，为刻画单圈图和双圈图中γ/γc值的界打下良好的基础。     

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.     

笛卡尔乘积图的成对控制

Let γpr（G） denote the paired domination number and G □ H denote the Cartesian product of graphs G and H. In this paper we show that for all graphs G and H without isolated vertex, γpr（G）γpr（H）≤ 7γpr （G □H）.     

Constructive characterizations of （γp, γ）- and （γp, γpr）-trees

Let G =(V,E) be a graph without isolated vertices.A set S  V is a domination set of G if every vertex in V -S is adjacent to a vertex in S,that is N[S] = V .The domination number of G,denoted by γ(G),is the minimum cardinality of a domination set of G.A set S  V is a paired-domination set of G if S is a domination set of G and the induced subgraph G[S]has a perfect matching.The paired-domination number,denoted by γpr(G),is defined to be the minimum cardinality of a paired-domination set S in G.A subset S  V is a power domination set of G if all vertices of V can be observed recursively by the following rules:(i) all vertices in N[S] are observed initially,and(ii) if an observed vertex u has all neighbors observed except one neighbor v,then v is observed(by u).The power domination number,denoted by γp(G),is the minimum cardinality of a power domination set of G.In this paper,the constructive characterizations for trees with γp = γ and γpr = γp are provided respectively.     

Two classes of edge domination in graphs

Let (, resp.) be the number of (local) signed edge domination of a graph G [B. Xu, On signed edge domination numbers of graphs, Discrete Math. 239 (2001) 179-189]. In this paper, we prove mainly that and hold for any graph G of order n(n?4), and pose several open problems and conjectures.     

关于图符号的边控制 (英)

设γ's(G)和γ'ι(G)分别表示图G的符号边和局部符号边控制数,本文主要证明了:对任何n阶图G(n≥4),均有γ's(G)≤[11/6n-1]和γ'ι(G)≤2n-4成立,并提出了若干问题和猜想．     

Constructive characterizations of （γp, γ）- and （γp, γpr）-trees

Let G = （V, E） be a graph without isolated vertices. A set S lohtain in V is a domination set of G if every vertex in V - S is adjacent to a vertex in S, that is N[S] = V. The domination number of G, denoted by γ（G）, is the minimum cardinality of a domination set of G. A set S lohtain in V is a paired-domination set of G if S is a domination set of G and the induced subgraph G[S] has a perfect matching. The paired-domination number, denoted by γpr（G）, is defined to be the minimum cardinality of a paired-domination set S in G. A subset S lohtain in V is a power domination set of G if all vertices of V can be observed recursively by the following rules： （i） all vertices in N[S] are observed initially, and （ii） if an observed vertex u has all neighbors observed except one neighbor v, then v is observed （by u）. The power domination number, denoted by γp（G）, is the minimum cardinality of a power domination set of G. In this paper, the constructive characterizations for trees with γp=γ and γpr = γp are provided respectively.     

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.