The Vulnerability of Some Networks including Cycles via Domination Parameters

Let G = (V(G), E(G)) 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 dominationnumbers are the 2-dominationnumber, 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, k-pyramids, and n-gon books, via domination parameters.Then, exact solutions of the domination parameters are obtained for the above-mentioned graphs.


Introduction
Graph theory has become one of the most powerful mathematical tools in the analysis and study of the architecture of a network.Networks are important structures and appear in many different applications and settings.The study of networks has become an important area of multidisciplinary research involving computer science, mathematics, chemistry, social sciences, informatics, and other theoretical and applied sciences [1][2][3].
It is known that communication systems are often exposed to failures and attacks.So robustness of the network topology is a key aspect in the design of computer networks.The stability of a communication network, composed of processing nodes and communication links, is of prime importance to network designers.As the network begins losing links or nodes, eventually there is a loss in its effectiveness [4].In the literature, various measures were defined to measure the robustness of network and a variety of graph theoretic parameters have been used to derive formulas to calculate network vulnerability.Graph vulnerability relates to the study of graph when some of its elements (vertices or edges) are removed.The measures of graph vulnerability are usually invariants that measure how the deletion of one or more network elements changes properties of the network [5,6].The best known measure of reliability of a graph is its connectivity.The vertex (edge) connectivity is defined to be the minimum number of vertices (edges) whose deletion results in a disconnected or trivial graph [7].Then the toughness [8], the integrity [9], the domination number [10,11], the bondage number [12,13], the 2-domination number [14], and the 2-bondage number [15] have been defined.Moreover, there are many graph theoretical parameters depending upon local damage for the graphs like the average lower independence number [16,17], the average lower domination number [17,18], the average connectivity [19], the average lower connectivity [20] and the average lower bondage number [6].The average parameters have been found to be more useful in some circumstance than the corresponding measures based on worst-case situation [6].
A natural way to model the topology of a communications network is as a graph consisting of vertices and edges.In this paper, we consider simple finite undirected graphs by ignoring any variation in the type of edges.Let  = ((), ()) be a simple undirected graph of order .We begin by recalling some standard definitions that we need throughout this paper.For any vertex The maximum and minimum degrees of a graph  are denoted by Δ() and (), respectively [11].The graph  is called r-regular graph if   (V) =  for every vertex V ∈ ().The vertex V is called isolated vertex if   (V) = 0.The null graph on -vertices consists of -isolated vertices with no edges.The join of graphs  and , denoted by  ∨ , is obtained from the disjoint union  +  by adding the edges {V | V ∈ (),  ∈ ()} [21].We will use ⌊⌋ and ⌈⌉ for the largest integer not larger than  and smallest integer not less than , respectively.
Cycles and various related graphs have been studied for many reasons.Fans, wheels, pyramids, bipyramids, and cycle books are among such graphs.The definitions of these graphs will be given in Sections 3.2, 3.3, and 3.4.
Our aim in this paper is to consider the computing of the average lower domination number (ALDN) and the average lower 2-domination number (AL2DN) of some networks including cycles.In Section 2, definitions and well-known basic results have been given for ALDN and AL2DN, respectively.In Section 3, ALDN and AL2DN of some networks including cycles, namely, fans, -pyramids, and -gon books, have been determined.

The Average Lower Domination Number Parameters and Basic Results
A set  ⊆ () is a dominating set if every vertex in ()− is adjacent to at least one vertex in .In 2004, Henning introduced the concept of average domination and average independence [17].The average lower domination number of a graph , denoted by  av (), is defined as  av () = (1/|()|) ∑ V∈()  V (), where the lower domination number, denoted by  V (), is the minimum cardinality of a dominating set of the graph  that contains the vertex V [17,18,22].In [23], the average lower 2domination number of a graph  was defined.The AL2DN is defined by  2av () = (1/|()|) ∑ V∈()  2V (), where the lower 2-domination number, denoted by  2V (), of the graph  relative to V is the minimum cardinality of 2-dominating set in the graph  that contains the vertex V.Moreover, Turaci showed that AL2DN is more sensitive than other vulnerability parameters, namely, connectivity, domination number, ALDN, and 2-domination number, in [23].

Fans
Definition 21 (see [21]).If one joins a vertex of   ( ≥ 3) to all other vertices, the resulting graph is called a fan (also known as a shell), denoted by   .For  = 3, we notice that  3 ≡  3 .Fans can be described by the join operation   =  −1 + V, where  ≥ 3.There is a vertex with ( − 1)-degree, namely, , in the graph   .
Theorem 22.Let   be a fan of order  and  ≥ 5; then Proof.The 2-dominating set is formed by two ways in   .
Proof.Because Δ(BP()) <  + 1, the domination number (BP()) is greater than 1.Let V 1 and V 2 be vertices whose degrees are , and let  be a dominating set.The dominating set  is formed by 3 cases.
, the set  is a dominating set.
Case 2. Let V 1 ∈  and let   be any vertex of   .Due to the fact that Then, the set  includes only vertices of   .So, we have || ≥ ⌈/3⌉.
Theorem 28.Let () be a bipyramid of order  + 2 and  ≥ 7; then Proof.When  V (BP()) is calculated for all vertices in the graph BP(), the vertices in two cases should be examined.Let V 1 and V 2 be vertices whose degrees are .
Proof.Let   be the vertices of   , where 1 ≤  ≤ , and let V  be vertices of the graph  2 , where 1 ≤  ≤ 2. There are 3 cases while forming 2-dominating set.
Theorem 36.Let (, ) be an -gon book and  ≥ 5; then Proof.Let  1 and  2 be two major vertices.We have three cases depending on the dominating set of (, ) that are including major vertices or not.
Case 3. Let  3 be a dominating set, and let { 1 ,  2 } ∉  3 .Since the set  3 includes only minor vertices, we have two subcases depending on .
Proof.The proof directly comes from Theorem 39.

Conclusion
In this study, a new defined graph theoretical parameter, namely, the average lower 2-domination number and the average lower domination number, has been studied for the network vulnerability.Additionally, the stability of popular interconnection networks including cycles has been studied and their domination numbers, 2-domination numbers, average lower domination numbers, and average lower domination 2-numbers, have been computed.These networks have been modeled with the fans, the -pyramids, and the -gon books.