Domination Number Of Graph Research Articles

Further results on outer independent triple Roman domination

An outer-independent triple Roman dominating function (OI[3]RDF) on a graph G = ( V , E ) is function f : V → { 0 , 1 , 2 , 3 , 4 } having the property that (i) if f ( v ) = 0 , then v must have either a neighbor assigned 4 or two neighbors one of which is assigned 3 and the other at least 2 or v has three neighbors all assigned 2; (ii) no two vertices assigned 0 are adjacent; (iii) if f ( v ) = 1 , then v must have either a neighbor assigned at least 3 or two neighbors assigned 2; (iv) if f ( v ) = 2 , then v must have one neighbor assigned at least 2. The weight of an OI[3]RDF is the sum of its function value over the whole set of vertices, and the outer-independent triple Roman domination number of G is the minimum weight of an OI[3]RDF on G. In this paper, we continue the study of outer-independent triple Roman domination number in graphs. First, we characterize all graphs with small or large outer-independent triple Roman domination number, and then we establish relationships with some other related parameters. Finally we present a sharp upper bound for the outer-independent triple Roman domination number of trees.

Connected 𝐷 - Eccentric Domination in Graphs

Objectives: To introduce connected -eccentric point set, connected -eccentric number, connected -eccentric dominating set, connected -eccentric domination number in a graph and related concepts. Methods: -distance in graphs are used to find the connected -eccentric number and connected -eccentric domination number in graphs. Findings: The new term connected -eccentric domination in graphs are used in varies field like to construct least number of cell phone tower in low cost and traffic signal also. Novelty: Using the idea -distance, eccentricity in a graphs, the connected -eccentric dominating set and its number in graphs are obtained. Some points, observation, bounds and theorem related to connected -eccentric domination set and its number in graphs are stated and proved. 2010 Mathematics Subject Classification: 05C12, 05C69 Keywords: D-eccentric node, D-eccentric node-set, D-eccentric dominating set, D-eccentric domination number, Connected D-eccentric dominating set, Connected D-eccentric domination number

The equitable non-split domination number of graphs

Finding a group of dominant servers is a necessary step towards optimising service provisioning to all clients in the complex world of server-client networks. This crucial decision rests on the non-split dominance number in the graph architecture. We present the idea of an equitable non-split domination number in graphs to address the possible problem of unequal task distribution among servers. When the vertices of a dominating set, Dens, have the same colour in an equitable colouring scheme, the set is said to be equitable. The induced subgraph ⟨V −D⟩ also maintains its connectivity. In particular, the application of equitable non-split domination numbers to graphs like the Triangular Grid, Antiprism, Wheel, and Complete Graphs is explored in this study.

An Upper Bound on the Total Roman { 2 } -domination Number of Graphs with Minimum Degree Two

A total Roman \(\{2\}\)-dominating function on a graph \(G = (V,E)\) is a function \(f:V\rightarrow\{0,1,2\}\) with the properties that (i) for every vertex \({v}\in V\) with \(f({v})=0\), \(f(N({v}))\ge2\) and (ii) the set of vertices with \(f({v})>0\) induces a subgraph with no isolated vertices. The weight of a total Roman \(\{2\}\)-dominating function is the value \(f(V)=\sum_{{v}\in V}f({v})\), and the minimum weight of a total Roman \(\{2\}\)-dominating function is called the total Roman \(\{2\}\)-domination number and denoted by \(\gamma_{tR2}(G)\). In this paper, we prove that for every graph \(G\) of order \(n\) with minimum degree at least two, \(\gamma_{tR2}({G})\leq \frac{5n}{6}\).

Domination Numbers in Graphs Resulting from Shackle Operations with Linkage of any Graph

The domination number is the number of dominating nodes in a graph that can dominate the surrounding connected nodes with a minimum number of dominating nodes. This domini number is denoted by γ(G). In this research, we will examine the domination number of the distance between two graphs resulting from the shackle operation with any graph as linkage. This differs from previous research, namely the domination of numbers at one and two distances. This study emphasizes how the results of operations on the shackle are connected to the shackle graph as any graph connects the copy. Any graph here means all graphs are connected and generally accepted. The method used in this research is pattern recognition and axiomatic deductive methods. The pattern detection method examines patterns where a graph's number of dominating points can dominate the connected points around it with a minimum number of dominating nodes. Meanwhile, axiomatic deductive is a research method that uses the principles of deductive proof that apply to mathematical logic by using existing axioms or theorems to solve a problem. The Result of graph S_n with t copies and S_m as linkage, then the two-distance domination number in the graph resulting from the shackle operation is γ_2 (Shack(S_n,S_m,t) )=t-1; graph S_n with t copies and C_m as linkage, then the two-distance domination number in the graph resulting from the shackle operation is γ_2 (Shack(S_n,C_m,t) )={■(t,for 3≤m≤6@⌈n/5⌉(t-1),for m≥7)┤; graph C_n with t copies and S_m as linkage, then the two-distance domination number in the graph resulting from the shackle operation isγ_2 (Shack(C_n,S_m,t) )={■(t-1,for n=3@t,for 4≤n≤5@⌈n/5⌉t,for n≥6)┤ This research provides benefits and adds to research results in the field of graph theory specialization of two-distance domination numbers in the result graph of shackle operation with linkage any graph.

Restrained captive domination number

Abstract The restrained captive domination number (RCDN), denoted by γ Rca ( G ) , {\gamma }_{{\rm{Rca}}}(G), is a new definition of domination number in graphs introduced in this article. If D D is a captive dominating set and G [ V − D ] G{[}V-D] has no isolated vertex, then D D is a restrained captive dominating set (RCDS) of graph G G . Furthermore, some properties, theorems, and propositions are determined. The inverse RCDN is introduced too. The RCDN in complement graphs is discussed. Finally, the RCDS of some graphs is determined. We have explained that if the graph has pendent vertex, then it has no RCDN.

3-component domination numbers in graphs

Let k be a positive integer and let G=(V(G),E(G)) be a graph. A vertex set D is a k-component dominating set of G if every vertex outside D in G has a neighbor in D and every component of the subgraph G[D] of G induced by D contains at least k vertices. The minimum cardinality of a k-component dominating set of G is the k-component domination number γk(G) of G. It was conjectured that if G is a connected graph of order n≥k+1, and minimum degree at least 2, then γk(G)≤2kn2k+3 except for a finite set of graphs. In this paper, we focus on the parameter γ3(G) of G. We first determine the exact values of 3-component domination numbers of paths and cycles. We then proceed to show that if G is a connected graph of order n with minimum degree at least 2 and maximum degree at most 3, then γ3(G)≤2n3, unless G is one of seven special graphs. This result provides positive support for the conjecture and also generalizes a result by Alvarado et al. (2016) [1].

On the strength and domination number of graphs

On the strength and domination number of graphs

Minimum Dominating Set for the Prism Graph Family

The dominating set of the graph G is a subset D of vertex set V, such that every vertex not in V-D is adjacent to at least one vertex in the vertex subset D. A dominating set D is a minimal dominating set if no proper subset of D is a dominating set. The number of elements in such set is called as domination number of graph and is denoted by $\gamma(G)$. In this work the domination numbers are obtained for family of prism graphs such as prism CL_n, antiprism Q_n and crossed prism R_n by identifying one of their minimum dominating set.

On the Domination Numbers of Certain Prism Graphs

A dominating set S of a graph , is a subset of the vertex set V (G) such that any vertex not in S is adjacent to at least one vertex in S .The domination number of a graph G denoted by is the minimum size of the dominating sets of G. In this paper we introduced the domination numbers of certain prism graphs.

A Note on Triple Repetition Sequence of Domination Number in Graphs

A set D subset of V(G) is a dominating set of a graph G if for all x ϵ V(G)\D, for some y ϵ D such that xy ϵ E(G). A dominating set D subset of V(G) is called a connected dominating set of a graph G if the subgraph <D> induced by D is connected. A connected domination number of G, denoted by γ_c(G), is the minimum cardinality of a connected dominating set D. The triple repetition sequence denoted by {S_n:n ϵ Z+} is a sequence of positive integers which is repeated thrice, i.e., {S_n}={1,1,1,2,2,2,3,3,3, ...}. In this paper, we construct a combinatorial explicit formula for the triple repetition sequence of connected domination numbers of a triangular grid graph.Keywords: connected domination number; triangular grid graph; triple repetition sequence. AbstrakSuatu himpunan D subhimpunan dari V(G) adalah himpunan pendominasi graf G apabila untuk semua x ϵ V(G)\D, untuk suatu y ϵ D sehingga xy ϵ E(G). Suatu himpunan pendominasi D subhimpunan dari V(G) dikatakan himpunan pendominasi terhubung dari graf G apabila subgraf <D> yang diinduksi oleh D terhubung. Suatu bilangan pendominasi dari G, dinotasikan dengan γ_c(G), adalah kardinalitas minimum dari himpunan pendominasi terhubung D. Barisan pengulangan rangkap tiga yang dinotasikan dengan {S_n:n ϵ Z+} adalah suatu barisan bilangan bulat positif yang setiap sukunya berulang tiga kali, yakni, {S_n}={1,1,1,2,2,2,3,3,3, ...}. Dalam paper ini dikonstruksi suatu rumus eksplisit kombinatorial untuk barisan pengulangan rangkap tiga dari bilangan pendominasi terhubung graf grid triangular.Kata Kunci: bilangan pendominasi terhubung; graf grid triangular; barisan pengulangan rangkap tiga. 2020MSC: 05C69

Nordhaus–Gaddum type inequalities on the total Italian domination number in graphs

Let G be a graph with vertex set V(G). A total Italian dominating function (TIDF) on a graph G is a function f : V(G) → {0, 1, 2} such that (i) every vertex v with f(v) = 0 is adjacent to a vertex u with f(u) = 2 or to two vertices w and z with f(w) = f(z) = 1, and (ii) every vertex v with f(v) ≥ 1 is adjacent to a vertex u with f(u) ≥ 1. The total Italian domination number γtI(G) on a graph G is the minimum weight of a total Italian dominating function. In this paper, we present Nordhaus–Gaddum type inequalities for the total Italian domination number.

Total 2-Rainbow Domination in Graphs

A total k-rainbow dominating function on a graph G=(V,E) is a function f:V(G)→2{1,2,…,k} such that (i) ∪u∈N(v)f(u)={1,2,…,k} for every vertex v with f(v)=∅, (ii) ∪u∈N(v)f(u)≠∅ for f(v)≠∅. The weight of a total 2-rainbow dominating function is denoted by ω(f)=∑v∈V(G)|f(v)|. The total k-rainbow domination number of G is the minimum weight of a total k-rainbow dominating function of G. The minimum total 2-rainbow domination problem (MT2RDP) is to find the total 2-rainbow domination number of the input graph. In this paper, we study the total 2-rainbow domination number of graphs. We prove that the MT2RDP is NP-complete for planar bipartite graphs, chordal bipartite graphs, undirected path graphs and split graphs. Then, a linear-time algorithm is proposed for computing the total k-rainbow domination number of trees. Finally, we study the difference in complexity between MT2RDP and the minimum 2-rainbow domination problem.

On the $ \{2\} $-domination number of graphs

<abstract><p>Let $ G $ be a nontrivial graph and $ k\geq 1 $ an integer. Given a vector of nonnegative integers $ w = (w_0, \ldots, w_k) $, a function $ f: V(G)\rightarrow \{0, \ldots, k\} $ is a $ w $-dominating function on $ G $ if $ f(N(v))\geq w_i $ for every $ v\in V(G) $ such that $ f(v) = i $. The $ w $-domination number of $ G $, denoted by $ \gamma_{w}(G) $, is the minimum weight $ \omega(f) = \sum_{v\in V(G)}f(v) $ among all $ w $-dominating functions on $ G $. In particular, the $ \{2\} $-domination number of a graph $ G $ is defined as $ \gamma_{\{2\}}(G) = \gamma_{(2, 1, 0)}(G) $. In this paper we continue with the study of the $ \{2\} $-domination number of graphs. In particular, we obtain new tight bounds on this parameter and provide closed formulas for some specific families of graphs.</p></abstract>

Determination of bulog regional sub-division in east java using connected domination number theory

Bulog is a state-owned public company engaged in food logistics. There are about 18 Bulog warehouses in East Java Province with an uneven distribution. Because the scope of Bulog’s business includes logistics or warehousing, surveys and pest eradication, provision of plastic bags, transportation business, food commodity trading and retail business, there must be good coordination between regional sub-divisions and warehouse complexes. In this study, the minimum regional sub-division recommendations in the East Java region were determined using the theory of connected domination number in graphs. The location of the warehouse complex is described as the vertex and adjacent warehouse complexes are connected by edges. In this research, we also determine the connected domination number of unicyclic graph and vertex amalgamation operations for several classes of graphs.

Maximal first Zagreb index of trees with given Roman domination number

<abstract><p>The first Zagreb index of graphs is defined to be the sum of squares of degrees of all the vertices of graphs. It drew a great deal of attention in the past half-century. In this paper, we study the relationship between the first Zagreb index and Roman domination number of graphs. More precisely, we characterize the trees with the maximum first Zagreb index among trees with given Roman domination number.</p></abstract>

New Bounds on the Double Total Domination Number of Graphs

Let G be a graph of minimum degree at least two. A set Dsubseteq V(G) is said to be a double total dominating set of G if |N(v)cap D|ge 2 for every vertex vin V(G). The minimum cardinality among all double total dominating sets of G is the double total domination number of G. In this article, we continue with the study of this parameter. In particular, we provide new bounds on the double total domination number in terms of other domination parameters. Some of our results are tight bounds that improve some well-known results.

Correction: Cabrera Martínez et al. On the Secure Total Domination Number of Graphs. Symmetry 2019, 11, 1165

The authors wish to make the following corrections on paper [...]

BILANGAN DOMINASI LOKASI PADA LINE PAN GRAPH DAN MIDDLE PAN GRAPH

Let G is a simple, connected, and undirected graph. A set D is a subset of V. If every node of V adjacent to at least one node D, then D is a dominating set of the graph G . One of the topics from dominating set is locating dominating set. Locating dominating set is dominating set on condition if every two vertices u,v elements of V-D satisfy the intersection of N(v) and D not equal to the intersection of N(u) and D with u not equal to v. The locating domination number of a graph G is the minimum cardinality of a locating dominating set in a graph G . In this study discussed the locating domination number on line pan graph (L(Tn,1)) and middle pan graph (M(Tn,1)). Locating domination number was obtained by finding dominating set from some graph. Then, does the dominating set meet the condition of locating dominating set? If it meets locating dominating set condition, then we can find the locating domination number of the graphs. In the last procedure, we get pattern location domination number of line pan graph and middle pan graph. The results of the study obtained the locating domination number of (L(Tn,1)) and (M(Tn,1)) consecutive are floor function of (2n/5)+1 and floor function of ((2n+1)/3)+1.

A new upper bound on the total domination number in graphs with minimum degree six

A total dominating set in a graph G is a set of vertices of G such that every vertex is adjacent to a vertex of the set. The total domination number γt(G) is the minimum cardinality of a dominating set in G. Thomassé and Yeo (2007) conjectured that if G is a graph on n vertices with minimum degree at least 5, then γt(G)≤411n. In this paper, it is shown that the Thomassé–Yeo conjecture holds with strict inequality if the minimum degree at least 6. More precisely, it is proven that if G is a graph of order n with δ(G)≥6, then γt(G)≤513814145n<411−12510n. This improves the best known upper bounds to date on the total domination number of a graph with minimum degree at least 6.