## 4 Malatya Dominating Set Algorithm (MDSA)
### The First Malatya Centrality Value  
The Malatya centrality algorithm uses node values that calculate the strength of the nodes of the given graph relative to each other and the result value is called Malatya centrality value. A basic principle used in this paper is at this aim.  
**Principle 1**.: _The strength of an entity should be given as the relative strength it has relative to the strength of its neighbors in its society._  
The strength of the nodes of the given graph can be calculated and Definition 3 is given for this purpose based on Principle 1. This definition consists of calculating the centrality value of each node and then multiplying the resulting value by the ratio of the active degree to the node degree. In the first step, the active node degree is equal to the node degree, therefore the ratio will be equal to one. Active node degree is the number of neighbors of a node that are white nodes (initially, all nodes of graph are regarded as white nodes that indicate where it is not yet clear whether nodes belong to the dominating set or belong to the neighbor set).  
**Definition 3**.: _Assume that \(G=(V,E)\) is a graph and \(\forall v_{i}\in V\), set \(V_{N}(v_{i})\) is the set of neighbors of the node \(vi\), \(\forall vi\in V\)._  
\[\Psi_{1}(v_{i})=\frac{d_{a}(v_{i})}{d(v_{i})}\left(\sum_{v_{i}\in V,v_{j}\in V _{N}(v_{i})}\frac{d(v_{i})}{d(v_{j})}\right) \tag{2}\]  
The obtained value \(\Psi_{1}\) is called the first Malatya centrality value of node \(v_{i}\) where \(d(v_{i})\) is the degree of node \(v_{i}\), and \(d_{a}(v_{i})\) is the active degree of \(v_{i}\) where active degree is the number of white neighbors of related node (the related node in Equation (2) is \(v_{i},1\leq i\leq n\)).  
The centrality value of each node is calculated by using the equation in 2, and this value is based on the logic of Principle 1. In this way, it concludes in the calculation of a node's strength relative to its neighbors (the calculation of a node's strength relative to its neighbor is inversely proportional to the degree of the relevant node). The algorithms can be constructed to find solutions for Maximum Independent Set and Minimum Node-Cover problems in the given graph by using the first Malatya centrality values. The steps to calculate the first Malatya centrality are given in Algorithm 1, with pseudo-codes.  
```
functionFirstMalatyaCentrality(A,D,DA) output:MC1/The first Malatya centrality values input:A,//Adjacency matrix D,//Degree vector DA//Activedegree vector
1 fori \(\leftarrow\) 1...n //(A is of sizes n \(\times\) n)
2L \(\leftarrow\) Neighbors(i)
3 forj in L
4 MC1(i) = MC1(i) + D(i)/D(j)
5 end
6 end
7 MC1 = MC1*DA//vectorinner product
8 fori \(\leftarrow\) 1...n
9 ifD(i) \(\
eq\) 0
10 MC1(i) = MC1(i)/D(i)
11 end
12 end
13 returnMC1
```  
**Algorithm 1** The First Malatya Centrality Algorithm.  
The impact area of the first Malatya centrality value on the graph is given in Figure 1. The centrality value of each node is calculated according to the nodes directly connected to it. The first Malatya centrality value shows the logic of calculating the relative strength of a node relative to its neighbors. Assuming we want to compute the first centrality value of node 'b', the nodes 'a', 'c', and dashed nodes should be taken into consideration since nodes one step away are taken into consideration for computing the first Malatya centrality values. All nodes are one step far away from the 'b' node, these nodes are regarded as in the first Malatya centrality influential area (region). The First Malatya centrality value of a node involves using the degrees of nodes one step away from that node.