Domination of Polynomial with Application
Authors: Alyaa A.  Alwan  
Published: February 2025

In this paper, .We .initiate the study of domination. polynomial , consider G=(V,E) be a simple, finite, and directed graph without. isolated. vertex .We present a study of the Iraqi national power grid 400kv. We extracted the poly dominance as well as the reliability. dominance by dividing it in to .several regions. The MATLAB program was used to calculate the shortest path for the network.
Introduction:    The study of  domination polynomial and domination reliability are very important concepts ..The domination polynomial of a graphs is a graph invariant that encodes information about the number of dominating. Set of various sizes in the graphs [29].The domination. Polynomial has been studies extensively to have various applications, such as in the analysis of communication network.[30]. Domination reliability is a measure .of the. reliability .of a graph or network, it is defined as the probability. that a randomly selected subset of vertices in the graph forms a dominating sets . [31],[32] .
Objectives: the reliability of the Iraqi super grid will be calculated .Calculation the reliability of figure 1 is NP-Hard, as the electrical conductors interfering with each other. Therefore, we relied on the reduction method by definition ""  to calculate the reliability of figure 1 ,where the parallel law is used for reduction and the series law for merging components with each other then use the law of reliability of parallele-seriese to compute the reliability of figure 1 .
Iraqi electrical power system is divided in to six subsystems R1 is Iraqi North Zone reliability index, R2 is Dyala- Anbar Zone reliability index, R3 is Baghdad North Zone reliability index, R4 is Iraqi Middle Zone reliability index and R6 is Iraqi South Zone reliability index
Methods: we want to look at the domination problem from the reliability point of view. We are interested in the probability that in a graph with random failing vertices or edges, a dominating set ( with some properties ) exists. In this section we only show some possible direction for further research . assume that the vertices of the graph are dominating with a given probability  and the edges are perfectly reliable.
Let  be a graph whose vertices fail randomly and independently with a given probability  for all . A failure in the context of domination means, that the vertex is not in the dominating set. In such a graph we are interested in the reliability that a dominating set exists. If we assume that , for all , and  , then we can define the domination reliability polynomial as
.
Remark: The domination polynomial and the domination reliability polynomial are equivalent.   
Results: We study this work, as domination reliability is may reveal some new insights in to system reliability.
Conclusions:  In this paper,
In this paper, .We .initiate the study of domination. polynomial , consider G=(V,E) be a simple, finite, and directed graph without. isolated. vertex .We present a study of the Iraqi national power grid 400kv. We extracted the poly dominance as well as the reliability. dominance by dividing it in to .several regions. The MATLAB program was used to calculate the shortest path for the network.
Introduction:    The study of  domination polynomial and domination reliability are very important concepts ..The domination polynomial of a graphs is a graph invariant that encodes information about the number of dominating. Set of various sizes in the graphs [29].The domination. Polynomial has been studies extensively to have various applications, such as in the analysis of communication network.[30]. Domination reliability is a measure .of the. reliability .of a graph or network, it is defined as the probability. that a randomly selected subset of vertices in the graph forms a dominating sets . [31],[32] .
Objectives: the reliability of the Iraqi super grid will be calculated .Calculation the reliability of figure 1 is NP-Hard, as the electrical conductors interfering with each other. Therefore, we relied on the reduction method by definition ""  to calculate the reliability of figure 1 ,where the parallel law is used for reduction and the series law for merging components with each other then use the law of reliability of parallele-seriese to compute the reliability of figure 1 .
Iraqi electrical power system is divided in to six subsystems R1 is Iraqi North Zone reliability index, R2 is Dyala- Anbar Zone reliability index, R3 is Baghdad North Zone reliability index, R4 is Iraqi Middle Zone reliability index and R6 is Iraqi South Zone reliability index
Methods: we want to look at the domination problem from the reliability point of view. We are interested in the probability that in a graph with random failing vertices or edges, a dominating set ( with some properties ) exists. In this section we only show some possible direction for further research . assume that the vertices of the graph are dominating with a given probability  and the edges are perfectly reliable.
Let  be a graph whose vertices fail randomly and independently with a given probability  for all . A failure in the context of domination means, that the vertex is not in the dominating set. In such a graph we are interested in the reliability that a dominating set exists. If we assume that , for all , and  , then we can define the domination reliability polynomial as
.
Remark: The domination polynomial and the domination reliability polynomial are equivalent.   
Results: We study this work, as domination reliability is may reveal some new insights in to system reliability.
Conclusions:  In this paper, .we introduced the Iraqi 400kv grid and .compute dominating polynomial of .this network also, determined the dominating reliability of Iraqi super grid , the computation of this network is NP- hard, that. divides the network to find the minimal path of Iraqi 400kv grid to compute the dominating .reliability .and dominating polynomial.We study this work, as domination reliability is may reveal some new .insights in .to system reliability.
.we introduced the Iraqi 400kv grid and .compute dominating polynomial of .this network also, determined the dominating reliability of Iraqi super grid , the computation of this network is NP- hard, that. divides the network to find the minimal path of Iraqi 400kv grid to compute the dominating .reliability .and dominating polynomial.We study this work, as domination reliability is may reveal some new .insights in .to system reliability.