Italian Domination of Strong Product of Two Paths

The domination problem of graphs is an important issue in the field of graph theory.This paper mainly considers the Italian domination number of the strong product between two paths.By constructing recursive Italian dominating functions,the upper bound of its Italian domination number is obtained,and then a partition method is proposed to prove its lower bound.Finally,this paper yields a sharp bound for the Italian domination number of the strong product of paths.

图的Domination染色

图 G 的一个 Domination 染色是使得图 G 的每个顶点 v 控制至少一个色类(可能是自身的色类), 并且每一个色类至少被 G 中一个顶点控制的一个正常染色。 图 G 的 Domination 色数是图 G 的 Domination 染色所需最小的颜色数目,用 χdd(G) 表示。 本文研究了图 G 的 Domination 色数与图 G 通过某种操作得到图 G"的 Domination 色数之间的关系。

Complete Cototal Roman Domination Number of a Graph for User Preference Identification in Social Media

Many graph domination applications can be expanded to achieve complete cototal domination.If every node in a dominating set is regarded as a record server for a PC organization,then each PC affiliated with the organization has direct access to a document server.It is occasionally reasonable to believe that this gateway will remain available even if one of the scrape servers fails.Because every PC has direct access to at least two documents’servers,a complete cototal dominating set provides the required adaptability to non-critical failure in such scenarios.In this paper,we presented a method for calculating a graph’s complete cototal roman domination number.We also examined the properties and determined the bounds for a graph’s complete cototal roman domination number,and its applications are presented.It has been observed that one’s interest fluctuate over time,therefore inferring them just from one’s own behaviour may be inconclusive.However,it may be able to deduce a user’s constant interest to some level if a user’s networking is also watched for similar or related actions.This research proposes a method that considers a user’s and his channel’s activity,as well as common tags,persons,and organizations from their social media posts in order to establish a solid foundation for the required conclusion.

An Upper Bound for Total Domination Number

Let G=(V, E) be a simple graph without an isolate. A subset T of V is a total dominating set of G if for any there exists at least one vertex such that .The total domination number γ1(G) of G is the minimum order of a total dominating set of G. This paper proves that if G is a connected graph with n≥3 vertices and minimum degree at least two.

On the Uphill Domination Polynomial of Graphs

A path π = [v1, v2, …, vk] in a graph G = (V, E) is an uphill path if deg(vi) ≤ deg(vi+1) for every 1 ≤ i ≤ k. A subset S ⊆ V(G) is an uphill dominating set if every vertex vi ∈ V(G) lies on an uphill path originating from some vertex in S. The uphill domination number of G is denoted by γup(G) and is the minimum cardinality of the uphill dominating set of G. In this paper, we introduce the uphill domination polynomial of a graph G. The uphill domination polynomial of a graph G of n vertices is the polynomial , where up(G, i) is the number of uphill dominating sets of size i in G, and γup(G) is the uphill domination number of G, we compute the uphill domination polynomial and its roots for some families of standard graphs. Also, UP(G, x) for some graph operations is obtained.

On the Injective Equitable Domination of Graphs

A dominating set D in a graph G is called an injective equitable dominating set (Inj-equitable dominating set) if for every , there exists such that u is adjacent to v and . The minimum cardinality of such a dominating set is denoted by and is called the Inj-equitable domination number of G. In this paper, we introduce the injective equitable domination of a graph and study its relation with other domination parameters. The minimal injective equitable dominating set, the injective equitable independence number , and the injective equitable domatic number are defined.

关于图的积的Domination数

关于图的积的Ｄｏｍｉｎａｔｉｏｎ数康丽英，单而芳（石家庄铁道学院基础部石家庄０５００４３）（石家庄师专数学系石家庄０５００４１）关键词：图；θ－图；Ｄｏｍｉｎａｔｉｎｇ集ＡＭＳ（１９９１）主题分类：０５Ｃ３５．本文所讨论的图均为无环、无重边的有限简单...

Lower Bounds on the Majority Domination Number of Graphs

Let G=(V,E) be a simple graph. For any real valued function f∶V→R and SV, let f(S)=∑ u∈S?f(u). A majority dominating function is a function f∶V→{-1,1} such that f(N)≥1 for at least half the vertices v∈V. Then majority domination number of a graph G is γ maj(G)=min{f(V)|f is a majority dominating function on G}. We obtain lower bounds on this parameter and generalize some results of Henning.

On the Total Domination Number of Graphs with Minimum Degree at Least Three

Let G be a simple graph with no isolated vertices. A set S of vertices of G is a total dominating set if every vertex of G is adjacent to some vertex in S . The total domination number of G , denoted by γ t (G) , is the minimum cardinality of a total dominating set of G . It is shown that if G is a graph of order n with minimum degree at least 3, then γ t (G)≤n/2 . Thus a conjecture of Favaron, Henning, Mynhart and Puech is settled in the affirmative.

Signed Roman (Total) Domination Numbers of Complete Bipartite Graphs and Wheels

A signed(res. signed total) Roman dominating function, SRDF(res.STRDF) for short, of a graph G =(V, E) is a function f : V → {-1, 1, 2} satisfying the conditions that(i)∑v∈N[v]f(v) ≥ 1(res.∑v∈N(v)f(v) ≥ 1) for any v ∈ V, where N [v] is the closed neighborhood and N(v) is the neighborhood of v, and(ii) every vertex v for which f(v) =-1 is adjacent to a vertex u for which f(u) = 2. The weight of a SRDF(res. STRDF) is the sum of its function values over all vertices.The signed(res. signed total) Roman domination number of G is the minimum weight among all signed(res. signed total) Roman dominating functions of G. In this paper,we compute the exact values of the signed(res. signed total) Roman domination numbers of complete bipartite graphs and wheels.

Bounds on Fractional Domination of Some Products of Graphs

Let γ f(G) and γ~t f(G) be the fractional domination number and fractional total domination number of a graph G respectively. Hare and Stewart gave some exact fractional domination number of P n×P m (grid graph) with small n and m . But for large n and m , it is difficult to decide the exact fractional domination number. Motivated by this, nearly sharp upper and lower bounds are given to the fractional domination number of grid graphs. Furthermore, upper and lower bounds on the fractional total domination number of strong direct product of graphs are given.

On Graphs with Equal Connected Domination and 2-connected Domination Numbers

A subset S of V is called a k-connected dominating set if S is a dominating set and the induced subgraph S has at most k components.The k-connected domination number γck（G） of G is the minimum cardinality taken over all minimal k-connected dominating sets of G.In this paper,we characterize trees and unicyclic graphs with equal connected domination and 2-connected domination numbers.

Independent Roman{2}-Domination in Trees

For a graph G=(V,E),a Roman{2}-dominating function f:V→{0,1,2}has the property that for every vertex v∈V with f(v)=0,either v is adjacent to at least one vertex u for which f(u)=2,or at least two vertices u1 and u2 for which f(u1)=f(u2)=1.A Roman{2}-dominating function f=(V0,V1,V2)is called independent if V1∪V2 is an independent set.The weight of an independent Roman{2}-dominating function f is the valueω(f)=Σv∈V f(v),and the independent Roman{2}-domination number i{R2}(G)is the minimum weight of an independent Roman{2}-dominating function on G.In this paper,we characterize all trees with i{R2}(T)=γ(T)+1,and give a linear time algorithm to compute the value of i{R2}(T)for any tree T.

On Total Domination Polynomials of Certain Graphs

We have introduced the total domination polynomial for any simple non isolated graph G in [7] and is defined by Dt（G, x） = ∑in=yt（G） dr（G, i） x＇, where dr（G, i） is the cardinality of total dominating sets of G of size i, and yt（G） is the total domination number of G. In [7] We have obtained some properties of Dt（G, x） and its coefficients. Also, we have calculated the total domination polynomials of complete graph, complete bipartite graph, join of two graphs and a graph consisting of disjoint components. In this paper, we presented for any two isomorphic graphs the total domination polynomials are same, but the converse is not true. Also, we proved that for any n vertex transitive graph of order n and for any v ∈ V（G）, dt（G, i） = 7 dt（V）（G, i）, 1 〈 i 〈 n. And, for any k-regular graph of order n, dr（G, i） = （7）, i 〉 n-k and d,（G, n-k） = （kn） - n. We have calculated the total domination polynomial of Petersen graph D,（P, x） = 10X4 ＋ 72x5 ＋ 140x6 ＋ 110x7 ＋ 45x8 ＋ [ 0x9 ＋ x10. Also, for any two vertices u and v of a k-regular graph Hwith N（u） ≠ N（v） and if Dr（G, x） = Dt（ H, x ）, then G is also a k-regular graph.

The Generalization of Signed Domination Number of Two Classes of Graphs

Let be a graph. A function is said to be a Signed Dominating Function (SDF) if holds for all . The signed domination number . In this paper, we determine the exact value of the Signed Domination Number of graphs and for , which is generalized the known results, respectively, where and are denotes the k-th power graphs of cycle and path .

On Minus Paired-Domination in Graphs

The study of minus paired domination of a graph G=(V,E) is initiated. Let SV be any paired dominating set of G , a minus paired dominating function is a function of the form f∶V→{-1,0,1} such that f(v)= 1 for v∈S, f(v)≤0 for v∈V-S , and f(N)≥1 for all v∈V . The weight of a minus paired dominating function f is w(f)=∑f(v) , over all vertices v∈V . The minus paired domination number of a graph G is γ - p( G )=min{ w(f)|f is a minus paired dominating function of G }. On the basis of the minus paired domination number of a graph G defined, some of its properties are discussed.

Signed Total Domination in Graphs

Let G=(V,E) be a simple graph. For any real valued function f:V →R, the weight of f is f(V) = ∑f(v) over all vertices v∈V . A signed total dominating function is a function f:V→{-1,1} such that f(N(v)) ≥1 for every vertex v∈V . The signed total domination number of a graph G equals the minimum weight of a signed total dominating function on G . In this paper, some properties of the signed total domination number of a graph G are discussed.

Signed total domination in nearly regular graphs

A function f： V（ G）→{1,1} defined on the vertices of a graph G is a signed total dominating function （STDF） if the sum of its function values over any open neighborhood is at least one. An STDF f is minimal if there does not extst a STDF g： V（G）→{-1,1}, f≠g, for which g （ v ）≤f（ v ） for every v∈V（ G ）. The weight of a STDF is the sum of its function values over all vertices. The signed total domination number of G is the minimum weight of a STDF of G, while the upper signed domination number of G is the maximum weight of a minimal STDF of G, In this paper, we present sharp upper bounds on the upper signed total domination number of a nearly regular graph.

CONVEX CONCENTRATION INEQUALITIES FOR CONTINUOUS GAS AND STOCHASTIC DOMINATION

In this article, we consider the continuous gas in a bounded domain ∧ of R^＋ or R^d described by a Gibbsian probability measure μη∧ associated with a pair interaction φ, the inverse temperature β, the activity z 〉 0, and the boundary condition η. Define F ∫ωf（s）wA（ds）. Applying the generalized Ito＇s formula for forward-backward martingales （see Klein et M. [5]）, we obtain convex concentration inequalities for F with respect to the Gibbs measure μη∧. On the other hand, by FKG inequality on the Poisson space, we also give a new simple argument for the stochastic domination for the Gibbs measure.

图的笛卡儿积的Domination数

对某些图类证明了 Vizing 猜想.设 G 为任意一个图,若图 H 满足下列条件之一,则 y(G×H)≥y(G)y(H):①图 H 为 n-圈 C_n;②y(H)≤2;③y(H)=n,H 中至少含有 n-2条互不邻接的端线.