Lower bounds for the domination number and the total domination number of direct product graphs

A sharp lower bound for the domination number and the total domination number of the direct product of finitely many complete graphs is given: . Sharpness is established in the case when the factors are large enough in comparison to the number of factors. The main result gives a lower bound for the domination (and the total domination) number of the direct product of two arbitrary graphs: γ(G×H)≥γ(G)+γ(H)−1. Infinite families of graphs that attain the bound are presented. For these graphs it also holds that γ_t(G×H)=γ(G)+γ(H)−1. Some additional parallels with the total domination number are made.     

On the total domination number of cross products of graphs

We give lower and upper bounds on the total domination number of the cross product of two graphs, γ_t(G×H). These bounds are in terms of the total domination number and the maximum degree of the factors and are best possible. We further investigate cross products involving paths and cycles. We determine the exact values of γ_t(G×Pn) and γ_t(C_n×C_m) where P_n and C_n denote, respectively, a path and a cycle of length n.     

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.     

Bounds relating the weakly connected domination number to the total domination number and the matching number

Let G=(V,E) be a connected graph. A dominating set S of G is a weakly connected dominating set of G if the subgraph (V,E∩(S×V)) of G with vertex set V that consists of all edges of G incident with at least one vertex of S is connected. The minimum cardinality of a weakly connected dominating set of G is the weakly connected domination number, denoted . 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 minimum cardinality of a total dominating set of G is the total domination number γ_t(G) of G. In this paper, we show that . Properties of connected graphs that achieve equality in these bounds are presented. We characterize bipartite graphs as well as the family of graphs of large girth that achieve equality in the lower bound, and we characterize the trees achieving equality in the upper bound. The number of edges in a maximum matching of G is called the matching number of G, denoted α^′(G). We also establish that , and show that for every tree T.     

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.     

An upper bound for domination number of 5-regular graphs

Let G = (V, E) be a simple graph. A subset S ⊆ V is a dominating set of G, if for any vertex u ∈ V-S, there exists a vertex v ∈ S such that uv ∈ E. The domination number, denoted by γ(G), is the minimum cardinality of a dominating set. In this paper we will prove that if G is a 5-regular graph, then γ(G) ⩽ 5/14n.     

Remarks on restrained domination and total restrained domination in graphs

The restrained domination number ^r(G) and the total restrained domination number _t ^r (G) of a graph G were introduced recently by various authors as certain variants of the domination number (G) of (G). A well-known numerical invariant of a graph is the domatic number d(G) which is in a certain way related (and may be called dual) to (G). The paper tries to define analogous concepts also for the restrained domination and the total restrained domination and discusses the sense of such new definitions.This research was supported by Grant MSM 245100303 of the Ministry of Education, Youth and Sports of the Czech Republic.     

A constructive characterization of total domination vertex critical graphs

Let G be a graph of order n and maximum degree Δ(G) and let γ_t(G) denote the minimum cardinality of a total dominating set of a graph G. A graph G with no isolated vertex is the total domination vertex critical if for any vertex v of G that is not adjacent to a vertex of degree one, the total domination number of G−v is less than the total domination number of G. We call these graphs γ_t-critical. For any γ_t-critical graph G, it can be shown that n≤Δ(G)(γ_t(G)−1)+1. In this paper, we prove that: Let G be a connected γ_t-critical graph of order n (n≥3), then n=Δ(G)(γ_t(G)−1)+1 if and only if G is regular and, for each v∈V(G), there is an A⊆V(G)−{v} such that N(v)∩A=0?, the subgraph induced by A is 1-regular, and every vertex in V(G)−A−{v} has exactly one neighbor in A.     

The ratio of the irredundance number and the domination number for block-cactus graphs

Let γ(G) and ir(G) denote the domination number and the irredundance number of a graph G, respectively. Allan and Laskar [Proc. 9th Southeast Conf. on Combin., Graph Theory & Comp. (1978) 43–56] and Bollobás and Cockayne [J. Graph Theory (1979) 241–249] proved independently that γ(G) < 2ir(G) for any graph G. For a tree T, Damaschke [Discrete Math. (1991) 101–104] obtained the sharper estimation 2γ(T) < 3ir(T). Extending Damaschke's result, Volkmann [Discrete Math. (1998) 221–228] proved that 2γ(G) ≤ 3ir(G) for any block graph G and for any graph G with cyclomatic number μ(G) ≤ 2. Volkmann also conjectured that 5γ(G) < 8ir(G) for any cactus graph. In this article we show that if G is a block-cactus graph having π(G) induced cycles of length 2 (mod 4), then γ(G)(5π(G) + 4) ≤ ir(G)(8π(G) + 6). This result implies the inequality 5γ(G) < 8ir(G) for a block-cactus graph G, thus proving the above conjecture. © 1998 John Wiley & Sons, Inc. J. Graph Theory 29: 139–149, 1998     

两类图的符号控制数

图G的符号控制数γs(G)有着许多重要的应用背景,因而确定其精确值有重要意义.Cm表示m个顶点的圈,n-Cm和n·Cm分别表示恰有一条公共边或一个公共顶点的n个Cm的拷贝.给出了n-Cm和n·Cm的符号控制数.     

Signed and minus domination in bipartite graphs

The paper studies the signed domination number and the minus domination number of the complete bipartite graph K _{p, q} .     

Roman domination in regular graphs

A Roman domination function on a graph G=(V(G),E(G)) is a function f:V(G)→{0,1,2} satisfying the condition that every vertex u for which f(u)=0 is adjacent to at least one vertex v for which f(v)=2. The weight of a Roman dominating function is the value f(V(G))=∑_u∈V(G)f(u). The minimum weight of a Roman dominating function on a graph G is called the Roman domination number of G. Cockayne et al. [E. J. Cockayne et al. Roman domination in graphs, Discrete Mathematics 278 (2004) 11-22] showed that γ(G)≤γ_R(G)≤2γ(G) and defined a graph G to be Roman if γ_R(G)=2γ(G). In this article, the authors gave several classes of Roman graphs: P_3k,P_3k+2,C_3k,C_3k+2 for k≥1, K_m,n for min{m,n}≠2, and any graph G with γ(G)=1; In this paper, we research on regular Roman graphs and prove that: (1) the circulant graphs and , n⁄≡1 (mod (2k+1)), (n≠2k) are Roman graphs, (2) the generalized Petersen graphs P(n,2k+1)( (mod 4) and ), P(n,1) (n⁄≡2 (mod 4)), P(n,3) ( (mod 4)) and P(11,3) are Roman graphs, and (3) the Cartesian product graphs are Roman graphs.     

关于图的强符号全控制数

图的强符号全控制数有着许多重要的应用背景,因而确定其下界有重要的意义.本文提出了图的强符号全控制数的概念,在构造适当点集的基础上对其进行了研究,给出了：（1）一般图的强符号全控制数的5个独立可达的下界及达到其界值的图;（2）确定了圈、轮图、完全图、完全二部图的强符号全控制数的值.     

THE TOTAL CHROMATIC NUMBER OF PSEUDO-OUTERPLANAR GRAPHS

A Planar graph g is called a ipseudo outerplanar graph if there is a subset v.∈V(G),[V.]=i,such that G-V. is an outerplanar graph in particular when G-V.is a forest ,g is called a i-pseudo-tree .in this paper.the following results are proved;(1)the conjecture on the total coloring is true for all 1-pseudo-outerplanar graphs;(2)X1(G) 1 fo any 1-pseudo outerplanar graph g with △(G)≥3,where x4(G)is the total chromatic number of a graph g.     

Properties of total domination edge-critical 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. The minimum cardinality of a total dominating set of G is the total domination number γ_t(G) of G. The graph G is total domination edge critical if for every edge e in the complement of G, γ_t(G+e)<γ_t(G). We call such graphs γ_tEC. Properties of γ_tEC graphs are established.     

Inequality of Nordhaus-Gaddum type for total outer-connected domination in graphs

A set S of vertices in a graph G = (V, E) without isolated vertices is a total outer-connected dominating set (TCDS) of G if S is a total dominating set of G and G[V − S] is connected. The total outer-connected domination number of G, denoted by γ _tc (G), is the minimum cardinality of a TCDS of G. For an arbitrary graph without isolated vertices, we obtain the upper and lower bounds on γ _tc (G) + γ _tc ($ \bar G $ \bar G ), and characterize the extremal graphs achieving these bounds.     

图的全控制数和匹配数的可比较性

设tγ(G)为G的全控制数.证明了:(1)对广义θ-图G,tγ(G)≤α(G) 1;(2)对任意k-正则无爪图G,k≥3,有tγ(G)≤α(G).这里α(G)表示G的匹配数.作为结果(2)的推论,对k-正则无爪图(k≥3),证明了Favaron猜想是成立的.即对最小度不小于3的简单图,有tγ(G)≤12 V(G).此外,举例说明了当图的最小度不超过2时,对一般图而言,匹配数与全控制数不可比较.