## 4 Malatya Dominating Set Algorithm (MDSA)
### The Second Malatya Centrality Value  
The previous section describes the method for calculating the first Malatya centrality value. The logic of calculating this value is based on the logic that the degree of a node can be computed according to the degrees of the neighbors of the corresponding node. The first Malatya centrality value reveals the status of a node relative to its neighbors. The obtained value is also called the relative strength of the corresponding node. Figure 1: The influential area of the first Malatya centrality values. The second stage centrality value, which is similar to the first Malatya centrality logic, can be calculated because a node's neighbors are also involved for the sake of the dominating set, not just its own neighbors. The second principle can be put forward based on this logic. **Principle 2. **_The relative strength of an entity can be given as the relative strength it has relative to the relative strength of its neighbors in its society._  
The second-level strength of the node can be obtained by using this principle, and Definition 4 is given for this reason. **Definition 4. **_Assume that \(G=(V,E)\) is a graph and \(\forall\
u_{i}\in V\) except selected nodes, \(V_{N}(v_{i})\) is the set of neighbors of node \(v_{i},\;v_{i}\in V\)._  
\[\Psi_{2}(v_{i})=\frac{1}{d(v_{i})}\frac{d_{a}(v_{i})}{d(v_{i})}\left(\sum_{v_{ i}\in V,v_{j}\in V_{N}(v_{i})}\frac{\Psi_{1}(v_{i})}{\Psi_{1}(v_{j})}\right) \tag{3}\]  
The obtained value \(\Psi_{2}\) is called the second Malatya centrality value of node \(\u u\) where \(d(v_{i})\) is the degree of node \(v_{i}\), and \(d_{a}(v_{i})\) is the active degree of \(\u v\) where active degree is the number of white neighbors of related node (the related node in Equation (3) is \(v_{i},1\leq i\leq n\)). The centrality value of each node is calculated with the correlation given in Definition 4, and this value is based on the logic of Principle 2. In this way, it reveals the calculation of the relative strength of a node compared to its neighbors. Initially, all nodes are designated as \"white\" nodes, and when a node is selected for the dominating set, the neighbors of that node are marked as \"grey\" nodes where a grey node does not belong to the dominating set and it has at least one neighbor node in the dominating set. This paper aims to calculate the second Malatya centrality value and obtain the MDS. Therefore, a two-stage centrality value or relative strength was calculated. The impact area (influential area) of the second Malatya centrality value is given in Figure 2. Since this paper aims to obtain the MDS in the given graph, the domain of the relative strength must be at least as shown in Figure 2. For example, when node 2 is included in the dominating set, node 3 is not included in the dominating set because it is adjacent to the selected node. To minimize the number of elements in the dominating set, node 4 is not included and node 5 may be included in the dominating set concerning the given graph in Figure 2. Algorithm 2 illustrates the pseudo-code of algorithm computing the second Malatya centrality values. Assume that the aim is to compute the second Malatya centrality value for node 'c'. The nodes 'b' and 'd' and the dashed nodes on the inner circle are taken into consideration for computing the first Malatya centrality value for node 'c'. This process is carried on  
Figure 2: The influential area of the second Malatya centrality values. for all nodes. The next step is to compute the second Malatya centrality value. All nodes two steps far away from node 'c' are taken into consideration for computing the second Malatya centrality value of node 'c' as seen in Equation (3) (Definition 4). To consider nodes two steps far away, the first Malatya centrality values should be used as similar to degree usage (as seen in Equation (2), Definition 1). To calculate the second Malatya centrality value of a node, the first Malatya centrality values of the neighbors of the related node are used.