Tufan Turaci

On the average lower bondage number of a graph

The domination number is an important subject that it has become one of the most widely studied topics in graph theory, and also is the most often studied property of vulnerability of communication networks. The vulnerability value of a communication network shows the resistance of the network after the disruption of some centers or connection lines until a communication breakdown. Let G = ( V ( G ) ,E ( G )) be a simple graph. The bondage number b ( G ) of a nonempty graph G is the smallest number of edges whose removal from G result in a graph with domination number greater than that of G . If we think a graph as a modeling of network, the average lower bondage number of a graph is a new measure of the graph vulnerability and it is defined by b av (G) = 1/| E(G) | ∑ e ∈ E(G) b e (G) , where the lower bondage number, denoted by b e ( G ), of the graph G relative to e is the minimum cardinality of bondage set in G that contains the edge e . In this paper, the above mentioned new parameter has been defined and examined. Then upper bounds, lower bounds and exact formulas have been obtained for any graph G . Finally, the exact values have been determined for some well-known graph families.

Relationships between vertex attack tolerance and other vulnerability parameters

Let G ( V,E ) be a simple undirected graph. Recently, the vertex attack tolerance (VAT) of G has been defined as τ( G ) = min {| S | / | V - S - C max ( G - S )|+1 : S ⊂ V } , where C max ( G − S ) is the order of a largest connected component in G − S . This parameter has been used to measure the vulnerability of networks. The vertex attack tolerance is the only measure that fully captures both the major bottlenecks of a network and the resulting component size distribution upon targeted node attacks. In this article, the relationships between the vertex attack tolerance and some other vulnerability parameters, namely connectivity, toughness, integrity, scattering number, tenacity, binding number and rupture degree have been determined.

The average lower reinforcement number of a graph

Let G = (V (G ),E (G )) be a simple undirected graph. The reinforcement number of a graph is a vulnerability parameter of a graph. We have investigated a refinement that involves the average lower reinforcement number of this parameter. The lower reinforcement number , denoted by r e ∗ (G ), is the minimum cardinality of reinforcement set in G that contains the edge e ∗ of the complement graph G . The average lower reinforcement number of G is defined by r av (G)=1/|E(G )| ∑e* * ∈ E (G ) r e* (G ) .In this paper, we define the average lower reinforcement number of a graph and we present the exact values for some well−known graph families.

The Vulnerability of Some Networks including Cycles via Domination Parameters

Let be an undirected simple connected graph. A network is usually represented by an undirected simple graph where vertices represent processors and edges represent links between processors. Finding the vulnerability values of communication networks modeled by graphs is important for network designers. The vulnerability value of a communication network shows the resistance of the network after the disruption of some centers or connection lines until a communication breakdown. The domination number and its variations are the most important vulnerability parameters for network vulnerability. Some variations of domination numbers are the 2-domination number, the bondage number, the reinforcement number, the average lower domination number, the average lower 2-domination number, and so forth. In this paper, we study the vulnerability of cycles and related graphs, namely, fans, -pyramids, and -gon books, via domination parameters. Then, exact solutions of the domination parameters are obtained for the above-mentioned graphs.