Abstract

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.

1. 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 [1–3].

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 open neighborhood of   is and closed neighborhood of   is . The degree of vertex   in is denoted by , that is, the size of its open neighborhood. The maximum degree of is and the minimum degree of is . The maximum and minimum degrees of a graph are denoted by and , respectively [11]. The graph is called r-regular graph if for every vertex . The vertex is called isolated vertex if . 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 [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.

2. 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 . The minimum cardinality taken over all dominating sets of is called the domination number of and it is denoted by [10]. Moreover, a 2-dominating set of a graph is a set of vertices of graph such that every vertex of has at least two neighbors in . The 2-domination number of a graph , denoted by , is the minimum cardinality of a 2-dominating set of the graph [2, 11, 14].

In 2004, Henning introduced the concept of average domination and average independence [17]. The average lower domination number of a graph , denoted by , is defined as , where the lower domination number, denoted by , is the minimum cardinality of a dominating set of the graph that contains the vertex [17, 18, 22]. In [23], the average lower 2-domination number of a graph was defined. The AL2DN is defined by , where the lower 2-domination number, denoted by , of the graph relative to is the minimum cardinality of 2-dominating set in the graph that contains the vertex . Moreover, Turaci showed that AL2DN is more sensitive than other vulnerability parameters, namely, connectivity, domination number, ALDN, and 2-domination number, in [23].

Theorem 1 (see [17]). For any graph G of order with domination number , , with equality if and only if G has a unique -set.

Theorem 2 (see [17]). If is a star graph of order , where , then .

Theorem 3 (see [17]). If is a path graph of order , then

Theorem 4 (see [17]). If is a complete graph of order , then .

Observation 5. If is a wheel graph of order , then .

Theorem 6 (see [15]). If is a complete graph of order , where , then .

Theorem 7 (see [15]). If is a path graph of order , then .

Theorem 8 (see [15]). If is a cycle graph of order , where , .

Theorem 9 (see [15]). If is a wheel graph of order , where ,

Theorem 10 (see [23]). Let be any connected graph of order . If -set is unique, then

Theorem 11 (see [23]). Let be any connected graph of order . If , then

Theorem 12 (see [23]). Let be any connected graph of order , where . Then,

Theorem 13 (see [23]). Let and be two connected graphs of order and , respectively. If and , then

Theorem 14 (see [23]). Let be any connected tree of order . If has s support vertices and leaf vertices, then .

Theorem 15 (see [23]). If is a path graph of order , then

Theorem 16 (see [23]). If is a wheel graph of order , where , then

Theorem 17 (see [23]). If is a complete graph of order , where , then .

Theorem 18 (see [23]). If is a star graph of order , where , then

3.1. Cycles

Theorem 19 (see [17]). If is a cycle graph of order , then .

Theorem 20 (see [23]). If is a cycle graph of order , then .

3.2. Fans

Definition 21 (see [21]). If one joins a vertex of to all other vertices, the resulting graph is called a fan (also known as a shell), denoted by . For , we notice that . Fans can be described by the join operation , where . There is a vertex with -degree, namely, , in the graph .

Theorem 22. Let be a fan of order and ; then .

Proof. The 2-dominating set is formed by two ways in .
Case  1. Let be a 2-dominating set and let include the vertex . So, the vertex dominates vertices of by once. Clearly, these -vertices which are dominated once form the path . Due to the fact that , these vertices must be taken to the set . So, is obtained.
Case  2. Let be a 2-dominating set and let not include vertex . So, the set must include vertices of subgraph path . By Theorem 7, we have .
By Cases  1 and 2, is obtained for . As a result, we get .

Theorem 23. Let be a fan of order and ; then .

Proof. By the definition of domination number and the structure of , the dominating set of is unique and . By Theorem 1, we get .

Theorem 24. Let be a fan of order and ; then

Proof. When is calculated for all vertices in , the vertices in three cases should be examined.
Case  1. For the vertex , the 2-dominating set must include the vertex by Theorem 22. The rest of the proof of this case is similar to Case  1 of Theorem 22. So, we get .
Case  2. For all vertices which forms the graph . We know that by the definition of ALDN. Furthermore, we know that the vertex must be in the 2-dominating set. So, we have for the sum of the lower 2-domination number of all vertices . As a result, is obtained.
By Cases  1 and 2, we get

Remark 25. Let be a fan of order and ; then

Proof. This is clear from Theorems 3 and 24.

3.3. -Pyramids

Definition 26 (see [21]). The join graph , where is the null graph of order , is called a -pyramid and is denoted by . The 2-pyramid is called bipyramid and is denoted by . The 1-pyramid is the wheel graph .

Theorem 27. Let be a bipyramid of order and ; then .

Proof. Because , the domination number is greater than 1. Let and be vertices whose degrees are , and let be a dominating set. The dominating set is formed by 3 cases.
Case  1. Let . Due to , the set is a dominating set.
Case  2. Let and let be any vertex of . Due to the fact that , the set is a dominating set.
Case  3. Let . Then, the set includes only vertices of . So, we have .
By Cases  1, 2, and 3, is obtained.

Theorem 28. Let be a bipyramid of order and ; then

Proof. When is calculated for all vertices in the graph , the vertices in two cases should be examined. Let and be vertices whose degrees are .
Case  1. For the vertices and , we know that the set dominates all vertices of . So, we get .
Case  2. For all vertices , by the definition of lower domination number and Case  2 of Theorem 27, we have for all vertices .
By Cases  1 and 2, we get .

Theorem 29. Let be a bipyramid of order and ; then . Furthermore, the 2-dominating set of is unique.

Proof. Let be the vertices of , where , and let be vertices of the graph , where . There are 3 cases while forming 2-dominating set.
Case  1. Let be a 2-dominating set of . Furthermore, the set contains vertices and . Clearly, . So, is obtained.
Case  2. Let be any 2-dominating set, and the set contains or . So, the set 1-dominates all vertices of . By the definition of , if we add vertices to the set , then all of graph’s vertices are 2-dominated. So, for . Hence, is obtained.
Case  3. Let be any 2-dominating set. Moreover, the set contains only vertices of . It is clear that the set is the 2-dominating set of . By Theorem 8, we have .
By Cases  1, 2, and 3 we get , , and for . As a result, by the definition of 2-domination number, we have and this 2-dominating set is unique.

Theorem 30. Let be a bipyramid of order and ; then

Proof. Since the 2-dominating set is unique, we have by Theorem 10.

Theorem 31. For the 3-pyramid with , .

Proof. There are 3 vertices whose degrees are in the graph and they are shown by , , and . We have four cases while 2-dominating set is forming.
Case  1. Let be any 2-dominating set of and let contain vertices , , and . It is clear that is obtained. So, the set is 2-dominating set of . Thus, is obtained.
Case  2. Let be any 2-dominating set of and let set include any two vertices of the vertices , , and . These two vertices are 2-dominated by the vertices of . If the remaining vertex is added to set, it is similar to set and all vertices of are 2-dominated. If the remaining vertex is not added to set, two vertices of must be added to the set . Thus, 3 is obtained.
Case  3. Let be any 2-dominating set of and let the set include any one vertex of , , and . So, the vertices of are dominated once by the set . Since the set does not include the remaining two vertices of the vertices , , and , vertices of must be taken to set to 2-dominate all vertices of and the remaining two vertices of the vertices , , and . Because , we have .
Case  4. Let be any 2-dominating set of and let the set not include any vertices of , , and . So, we must add vertices of the graph to set . We know that by Theorem 8. Thus, is obtained.
By Cases  1, 2, 3, and 4 we get , , , and for . As a result we have by the definition of 2-domination number and this 2-dominating set is unique.

Theorem 32. For the 3-pyramid with , .

Proof. Since the 2-dominating set is unique, we have by Theorem 10.

Theorem 33. Let be a -pyramid with and ; then .

Proof. The vertices of are denoted by , where , and the remaining -vertices are denoted by , where . We have three cases while 2-dominating set is forming.
Case  1. Let be any 2-dominating set of the graph and let contain only vertices , where . It is clear that set 2-dominated all vertices of . So, and are obtained.
Case  2. Let be any 2-dominating set of and let contain only vertices of , where . By Theorem 8, we have . As a result, is obtained.
Case  3. Let be any 2-dominating set of and let contain any vertex of and , where and . Let . Let . The vertex dominates vertices of by once and the vertex dominates vertices of by once. If any vertex of and any vertex of are added to set , then will be 2-dominating set of the graph . Hence, is obtained.
By Cases  1, 2, and 3 we get , , and for . As a result, by the definition of 2-domination number, we have .

Theorem 34. Let be a -pyramid with and ; then .

Proof. By Theorem 33, we have the lower 2-domination number as 4 for all vertices of . Thus, is obtained.

3.4. -Gon Books

Definition 35 (see [21]). When copies of share a common edge, they will form an -gon book of   pages and are denoted by . The degree set of is . Therefore, the vertices of are of two kinds: vertices of degree 2, which will be referred to as minor vertices, and vertices of degree , which will be referred to as major vertices. The minor vertices of are labeled , that is , where and .

Theorem 36. Let be an -gon book and ; then

Proof. Let and be two major vertices. We have three cases depending on the dominating set of that are including major vertices or not.
Case  1. Let be a dominating set, and let include two major vertices. The set dominates -vertices from all . So, there are -vertices which are not dominated from each . Then the -distinct path is obtained by these minor vertices. Thus, is obtained.
Case  2. Let be a dominating set, and let (or ). The vertex dominates the vertex and -vertices which are adjacent to the vertex from all . Then the remaining -vertices are not dominated from each . So, the -distinct path is obtained by these minor vertices. As a result, is obtained.
Case  3. Let be a dominating set, and let . Since the set includes only minor vertices, we have two subcases depending on .
Case  3.1. Let . Due to the structure of , the set must include -vertices from each . So, the whole vertices of are dominated. As a result we have .
Case  3.2. Let . Due to the structure of , two minor vertices which are neighbors to the major vertices must be taken to the set from any graph . Furthermore, the remaining -vertices are not dominated in the graph . Since -vertices must be taken to the set , is obtained.
By Cases  3.1 and 3.2, we getClearly, if , then we have , if , then we have , and if , then we have . As a result, we getThus the proof is completed.

Theorem 37. Let be an -gon book and ; then

Proof. While finding lower domination number of all vertices of , we have three cases depending on .
Case  1. Let , and let be a dominating set. Clearly, the set is a unique dominating set. By Theorem 10, we have Case  2. Let , and let be a dominating set. By Theorem 36, the set is formed by two ways. Clearly, is obtained for every . As a result, we have .
Case  3. Let , and let be a dominating set. By Case  2 of Theorem 36, we know that the set includes either the vertex or the vertex . Let . Clearly, the set contains vertices of by Theorem 36. Thus, we have for every . Furthermore, we have for every . Thus, By Cases  1, 2, and 3 the proof is completed.

Theorem 38. Let be an -gon book and ; then

Proof. Let and be two major vertices. We have two cases depending on .
Case  1. Let be even, and let be 2-dominating set. We know that the set contains either the vertex or the vertex by the definition of and Theorem 8. Thus, -vertices of each must be taken to the set . Since there is -graph , we have . So, is obtained.
Case  2. Let be odd. We have two subcases depending on the 2-dominating set of that are including major vertices or not.
Case  2.1. Let be 2-dominating set, and let include any major vertex. Clearly, the set must include two major vertices in this subcase by Theorem 8. So, the set dominates -vertices from all . Then, -vertices are not dominated from each . Thus, the -distinct path is obtained by these minor vertices. As a result, is obtained.
Case  2.2. Let be 2-dominating set, and let . Clearly, the set includes -vertices of of each by Theorem 7. Thus, is obtained.
By Cases  2.1 and 2.2 for and . Then we have for and . Furthermore, 2-dominating set is unique for and . As a result, is obtained.
By Cases  1 and 2, the proof is completed.

Theorem 39. Let be an -gon book, where and ; then

Proof. Let and be two major vertices. We have two cases depending on .
Case  1. Let be even. We have that the 2-dominating set including the vertex is 2-dominating set of any . Since the vertex is common vertex of each and by Theorem 8, we get . Similarly, we get . Let be minor vertex which is neighbor to the vertex of any , and let be minor vertex which is neighbor to the vertex in the same graph . It is easy to see that the 2-dominating set including the vertex is the 2-dominating set including the vertex by Theorem 8. Similarly, the 2-dominating set including the vertex is the 2-dominating set including the vertex . Thus, is obtained. Furthermore, we get for every , where and . As a result, is obtained.
Case  2. Let be odd. We have two subcases depending on .
Case  2.1. Let . We know that 2-dominating set is formed by two ways in Case  2 of Theorem 38. Clearly, is obtained for every vertex . As a result, is obtained.
Case  2.2. Let . We know that 2-dominating set is unique by Case  2 of Theorem 38. So, is obtained by Theorem 10.
By Cases  1 and 2 the proof is completed.

Theorem 40. Let be an -gon book, where and ; then

Proof. The proof directly comes from Theorem 39.

4. 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.

Competing Interests

The authors declare that they have no competing interests.

Acknowledgments

The authors would like to thank Professor Dr. Manlio Gaudioso for the constructive suggestions and comments that improve the quality of this paper.