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    Combining the Concepts of Residual and Domination in Graphs
    (Ios Press, 2019) Turaci, Tufan; Aytac, Aysun
    Let G = (V (G), E(G)) be a simple undirected graph. The domination and average lower domination numbers are vulnerability parameters of a graph. We have investigated a refinement that involves the residual domination and average lower residual domination numbers of these parameters. The lower residual domination number, denoted by gamma(R)(uk)(G), is the minimum cardinality of dominating set in G that received from the graph G where the vertex v(k) and all links of the vertex v(k) are deleted. The residual domination number of graphs G is defined as gamma(R)(G) = minv(k)is an element of V(G){gamma(R)(vk)(G)} . The average lower residual domination number of G is de- fined by gamma(R)(av)(G) = 1/vertical bar V(G)vertical bar Sigma(vk is an element of V(G)) gamma(R)(vk)(G). In this paper, we define the residual domination and the average lower residual domination numbers of a graph and we present the exact values, upper and lower bounds for some graph families.
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    ON THE DOMINATION, STRONG AND WEAK DOMINATION IN TRANSFORMATION GRAPH Gxy-
    (Util Math Publ Inc, 2019) Aytac, Aysun; Turaci, Tufan
    Let G = (V(G), E(G)) be a simple undirected graph of order vs and size m, and x, y, z be three variables taking value + or -. The transformation graph of G, G(xyz) is a simple graph having V(G) boolean OR E(G) as the vertex set. If an expression has n variables, and each variable can have the value + or -, the number of different combinations of values of the variables is 2(n). Thus, we obtain eight kinds of transformation graphs. A set S subset of V(G) is a dominating set if every vertex in V(G) - S is adjacent to at least one vertex in S. The minimum cardinality taken over all dominating sets of G is called the domination number of G and is denoted by gamma(G). There are several types of domination parameters depending upon the nature of dominating sets. In this paper, we investigate the domination number, the strong domination number and the weak domination number of the transformation graph G(xy)(-).
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    Vulnerability Measures of Transformation Graph Gxy+
    (World Scientific Publ Co Pte Ltd, 2015) Aytac, Aysun; Turaci, Tufan
    Several factors have to be taken into account in the design of large interconnection networks. Optimal design is important both to achieve good performance and to reduce the cost of construction and maintenance. Practical communication networks are exposed to failures of network components. Failures between nodes and connections happen and it is desirable that a network is robust in the sense that a limited number of failures does not break down the whole system. Robustness of the network topology is a key aspect in the design of computer networks. A variety of measures have been proposed in the literature to quantify the robustness of networks and a number of graph-theoretic parameters have been used to derive formulas for calculating network reliability. In this paper, we study the vulnerability of interconnection networks to the failure of individual nodes, using a graph-theoretic concept of domination and strong-weak domination numbers of the transformation graph G(xy)+ as a measure of network robustness.