Domination Number

Open Access Article

1 - Co-Roman domination in trees
Rana Khoeilar Marzieh sorudi

Abstract: Let G=(V,E) be a graph and let f:V(G)→{0,1,2} be a function&lrm;. &lrm;A vertex v is protected with respect to f&lrm;, &lrm;if f(v)>0 or f(v)=0 and v is adjacent to a vertex of positive weight&lrm;. &lrm;The function f is a co-Roman dominating function More

Abstract: Let G=(V,E) be a graph and let f:V(G)→{0,1,2} be a function&lrm;. &lrm;A vertex v is protected with respect to f&lrm;, &lrm;if f(v)>0 or f(v)=0 and v is adjacent to a vertex of positive weight&lrm;. &lrm;The function f is a co-Roman dominating function&lrm;, &lrm;abbreviated CRDF if&lrm;: &lrm;(i) every vertex in V is protected&lrm;, &lrm;and (ii) each u∈V with positive weight has a neighbor v∈V with f(v)=0 such that the function f_uv:V→{0,1,2}&lrm;, &lrm;defined by f_uv (v)=1&lrm;, &lrm;f_uv (u)=f(u)-1 and f_uv (x)=f(x)for x∈V-\{v,u}&lrm;, &lrm;has no unprotected vertex&lrm;. &lrm;The weight of f is ω(f)=∑_(v∈V)▒〖f(v)〗&lrm;. &lrm;The co-Roman domination number of a graph G &lrm;, &lrm;denoted by γ_cr G)&lrm;, &lrm;is the minimum weight of a co-Roman dominating function on G &lrm;. &lrm;In this paper, we first present an upper bound on the co-Roman domination number of trees in terms of order, the number of leaves and supports&lrm;. Then we find bounds on the co-Roman domination number of a graph and its other dominating parameters . Manuscript profile

Open Access Article

2 - Improvment Upper Bound of the Independence Nnmber of Maximum Independent Set in Unit Disk Graph
Gholam Hassan Shirdel mehdi jalinousi

10.30495/jnrm.2022.58031.2028

In a unit disk graph two vertices are adjacent if the distance between them is less than or equal to one with a two-dimensional Euclidean meter. The size of the maximal independent set in a graph G is called the independent number denoted by α(G). The size of the More

In a unit disk graph two vertices are adjacent if the distance between them is less than or equal to one with a two-dimensional Euclidean meter. The size of the maximal independent set in a graph G is called the independent number denoted by α(G). The size of the minimal connected dominating set in a graph G is called the connected domination number denoted by γ_c^((G)). A subset S of vertices in a graph is called a dominating set if every vertex is either in the subset or adjacent to a vertex in the S. A dominating set is connected if it induces a connected subgraph. A connected dominating set is often used as a virtual backbone in wireless sensor networks to improve communication and storage performance. Clearly the smaller virtual backbone gives the better performance.However computing a minimal connected dominating set is NP-hard. In other hand relation between the size of the minimal connected dominating set in a graph G is very important. The aim of this paper is to determine two better upper bounds of the independence number dependent on the connected domination number for a unit disk graph. Further we improve the upper bound to obtain the best bound with respect to the upper bounds obtained thus far. Manuscript profile

Open Access Article

3 - A Characterization of Trees with Large Roman Domination Number
حسین . Abdollahzadeh Ahangar مهلا . Khaibari N. Jafari Rad

A Roman dominating function (RDF) on a graph G = 􀵫V،E􀵯 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 onevertex v for which f(v) = 2. The weight of an RDF f is w(f) = Σ􀭴∈􀭚 f(v). More

A Roman dominating function (RDF) on a graph G = 􀵫V،E􀵯 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 onevertex v for which f(v) = 2. The weight of an RDF f is w(f) = Σ􀭴∈􀭚 f(v). The Romandomination number of G is the minimum weight of an RDF in G. In this paper, wecharacterize all trees T of order n whose Roman domination number is n − 3. Manuscript profile

Open Access Article

4 - Some Results on the Maximal 2-Rainbow Domination Number in Graphs
H. Abdollahzadeh Ahangar Z. Ghandali

A 2-rainbow dominating function ( ) of a graph is a function from the vertex set to the set of all subsets of the set such that for any vertex with the condition is fulfilled, where is the open neighborhood of . A maximal 2-rainbow dominating function on a graph is a 2- More

A 2-rainbow dominating function ( ) of a graph is a function from the vertex set to the set of all subsets of the set such that for any vertex with the condition is fulfilled, where is the open neighborhood of . A maximal 2-rainbow dominating function on a graph is a 2-rainbow dominating function such that the set is not a dominating set of . The weight of a maximal is the value . The maximal 2-rainbow domination number of a graph , denoted by , is the the minimum weight of a maximal of . In this paper, we continue the study of maximal 2-rainbow domination number. We characterize all graphs of order whose maximal 2-rainbow domination number is equal to 2 or 3. Finally, we characterize all graphs of order with for which . Manuscript profile

Open Access Article