#### Abstract

A graph with specified target vertices in vertex set is a -terminal graph. The -terminal reliability is the connection probability of the fixed target vertices in a -terminal graph when every edge of this graph survives independently with probability . For the class of two-terminal graphs with a large number of edges, Betrand, Goff, Graves, and Sun constructed a locally most reliable two-terminal graph for close to 1 and illustrated by a counterexample that this locally most reliable graph is not the uniformly most reliable two-terminal graph. At the same time, they also determined that there is a uniformly most reliable two-terminal graph in the class obtained by deleting an edge from the complete graph with two target vertices. This article focuses on the uniformly most reliable three-terminal graph of dense graphs with vertices and edges. First, we give the locally most reliable three-terminal graphs of and in certain ranges for close to 0 and 1. Then, it is proved that there is no uniformly most reliable three-terminal graph with specific and , where and . Finally, some uniformly most reliable graphs are given for vertices and edges, where and or and .

#### 1. Introduction

Network reliability is a hot topic which has been generally investigated using graph theoretic models. Network with vertices and edges can be modeled as a graph with the same number of vertices, edges, and interconnections as the network. For all-terminal reliability (connection probability of all vertices of a graph), lots of authors investigate the existence of a uniformly most reliable (all-terminal) graph for various values of and [1–8]. However, the research on -terminal reliability (connection probability of target vertices in a graph, where ) is mainly about the algorithm of computing the -terminal reliability polynomial [9–13], but only a few results on the construction of the uniformly most reliable -terminal graph.

In [8], Kelmans has shown that for , the uniformly most reliable graph is a complete graph with a matching removed (the matching of a graph is a set of edges with no common vertices between each other). In fact, the design of the real network often only needs to ensure the connectivity of critical vertices (target vertices). Therefore, the construction of the most reliable -terminal graph has high application value. There are a few studies on the construction of the most reliable -terminal structure. In [14], Betrand et al. proved that there is no uniformly most reliable two-terminal graph when . At the same time, they also proved that when , the uniformly most reliable two-terminal graph is a complete graph with removing an edge between nontarget vertices. It is natural to consider the following problems.

Problems: for the three-terminal graphs with a large number of edges, is there a uniformly most reliable graph? If it exists, what is its construction? If it does not exist, can we construct the locally most reliable three-terminal graph?

With these questions, we further study the existence of uniformly most reliable three-terminal graphs for large . It is difficult to find the exact cases that three target vertices are connected, which is NP-complete [15]. In this paper, we just consider the uniformly most reliable three-terminal dense graphs. In Section 2, some related basic definitions and notations are given. In Section 3, the locally most reliable three-terminal graphs for given are determined. We show that there is no uniformly most reliable three-terminal graph for vertices and edges where and and give the uniformly most reliable graphs for and . In Section 4, a uniformly most reliable three-terminal graph with vertices and edges is determined. The results are summarized in Section 5.

#### 2. Basic Concepts and Notations

For notations and terminologies not defined here, we refer to [16]. Let denote the number of subgraphs which is isomorphic to in . The complement of , denoted by , is the graph obtained by deleting the edges of and adding edges between all nonadjacent vertices in . If , then denotes the addition of the edge to , and denotes the deletion of the edge from . The union of simple graphs and , denoted by , is the graph with vertex set and edge set . Let be the path with vertices, be the complete graph with vertices, and be a star with vertices and edges.

A graph with three specified target vertices , and in is a three-terminal graph. Using denotes the set of all simple three-terminal graphs with vertices and edges. The connectivity probability of the three specified target vertices in graph when each edge of survives independently with a fixed probability is called the three-terminal reliability of (or the three-terminal reliability polynomial of ), denoted by . A -subgraph is a subgraph of in which vertices are connected in the subgraph. In particular, if the -subgraph with edges does not contain any -subgraph with less than edges, then it is minimal; otherwise, it is nonminimal. Clearly, -subgraph is a subgraph of in which three target vertices are connected in the subgraph. Similar to the definition of two-terminal reliability [14], the three-terminal reliability polynomial of the graph can be written aswhere (or simply ) is the number of -subgraphs of graph with edges.

Similar as the definitions of the uniformly most reliable two-terminal graph [14] and the locally most reliable all-terminal graph [3], we defined the uniformly most reliable graph and the locally most reliable graph for three-terminal graphs.

*Definition 1. *A graph is the uniformly most reliable graph in , if for all and all . In particular, for (or 1), if there is an such that for all and for all , then is the locally most reliable graph in for close to (or for close to ).

*Example 1. *Figure 1 shows all types of simple three-terminal graph in with three target vertices . Each edge of these graphs survives independently with probability .

In , the -subgraphs with 2 edges are , , and the -subgraphs with 3 edges are , , , and , while the -subgraph with 4 edges is . Obviously, , , and . Similarly, we can calculate , , which are , , and ; , , and ; and , , and . Figure 2 shows a visualization of reliability polynomials for all graphs in . Clearly, for all , , so is the uniformly most reliable graph in .

**(a)**

**(b)**

**(c)**

**(d)**

*Example 2. *Figure 3 shows two special simple three-terminal graphs in with three target vertices . Each edge of these graphs survives independently with probability . By calculation, we give a plot of as shown in Figure 4. Clearly, for close to 1, and for close to 0. This article later proves that is the locally most reliable graph for close to 1 and is the locally most reliable graph for close to 0 in .

Many studies focus on determining a uniformly most reliable graph for given number of vertices and edges , as shown in [1, 2, 14]. If there is no uniformly most reliable graph, researchers usually focus on determining the locally most reliable graph for close to 0 or 1, as shown in [3, 17]. At present, there are few studies on determining whether there is a uniformly most reliable three-terminal graph. Therefore, in this paper, we study the uniformly most reliable graph and the locally most reliable graph of dense three-terminal graph. One can see that some dense graphs have uniformly most reliable three-terminal graphs as Example 1, and some does not have the uniformly most reliable graph but have locally most reliable three-terminal graphs as Example 2.

**(a)**

**(b)**

#### 3. Some Locally Most Reliable Three-Terminal Graphs

In this section, the locally most reliable three-terminal graph with and for close to 0 is determined and the locally most reliable three-terminal graph with and for close to 1 is also determined. Then, it is shown that for and , there is no uniformly most reliable graph in . For and , there is a uniformly most reliable three-terminal graph. To prove these results, we first introduce some related lemmas.

Lemma 1. *(see [17]). Let and be positive integers. If , then the unique simple graph on vertices and edges with the maximum number of is . If , then there are two simple graphs with the maximum number of : and .*

In general, it is difficult to calculate the three-terminal reliability polynomial of graph. Therefore, we study the locally most reliable graph by the following Lemma 2, which is extracted from [14].

Lemma 2. *Let the three-terminal reliable polynomials of be and .*

Let for and for , where and are integers. Then,(1)for close to 0, if , then ;(2)for close to 1, if , then .

By Lemma 2, we get the following conclusions:(1)If is the locally most reliable graph for close to 0, then it must contain the triangle and the value of is the maximum among graphs containing the triangle in .(2)An -cutset of is a set of edges whose deletion results in the disconnection of vertices , and in and the number of edges is its size. The edge connectivity of , and in is the smallest size of an -cutset of , denoted by or simply . If is the locally most reliable graph for close to 1, then it must have the maximum . Since () and , where is the number of the -cutsets with size , must have the minimum among the graphs with the largest edge connectivity of , and in .

Now, we demonstrate the locally most reliable graph for three-terminal graphs for close to 0 or 1. We first introduce two important graphs for this section, as follows:

Let and be positive integers. Using denotes the three-terminal graph on vertices and edges with vertex set and edge set .

Let and be positive integers. Using denotes the three-terminal graph on vertices and edges with vertex set and edge set .

Figure 5 depicts these two three-terminal graphs with 11 vertices and 51 edges.

**(a)**

**(b)**

Theorem 1. *Let , and be positive integers. Then,*(1)*if , then the graph is the unique locally most reliable graph in for close to 0;*(2)*if , then the graph is the unique locally most reliable graph in for close to 0.*

*Proof. *Suppose that is the locally most reliable graph in for close to 0. Then, by Lemma 2, must contain the triangle and (the number of -subgraphs with 3 edges in ) must be maximum among graphs containing the triangle in .

For the sake of calculating , using denotes the number of -subgraphs with 3 edges containing 2,1,0 edges in triangle , respectively. Then, is the number of sets , , and (), is the number of sets , , , , , and (), and is the number of set (). It is no hard to see that . Clearly, for graphs containing the triangle in , is a constant which equals , and takes the maximum value if and only if and attains the maximum value. Note that if takes the maximum value , then the value of also reaches its maximum; that is, contains the edges for all . Now, consider the remaining edges among nontarget vertices in that has not been described. Since is a dense graph, it is often easier to consider the deleted edges among nontarget vertices.

By Lemma 2, we continue to calculate the coefficients (), where and are the number of minimal -subgraphs and nonminimal -subgraphs with edges, respectively. We now compute . Clearly, is the sum of the numbers of sets , , and (). The nonminimal -subgraph with 4 edges includes two cases: one is containing the minimal -subgraph with 2 edges and the other is containing the minimal -subgraph with 3 edges but no minimal -subgraph with 2 edges. By calculation, and which are constants. So, is a constant.

Furthermore, we need to calculate . Clearly, is the sum of the numbers of sets , , , and (). The nonminimal -subgraph with 5 edges includes three cases: containing the minimal -subgraph with 2 edges, containing the minimal -subgraph with 3 edges but no minimal -subgraph with 2 edges, and containing the minimal -subgraph with 4 edges but no minimal -subgraph with less than 4 edges. By calculation, , which is a constant. Andwhere is the degree of in , and is the number of edges deleted on the nontarget vertex , which is the degree of in . Note that the number of subgraphs as in is . Obviously, of is related to , the number of subgraphs as in . There are at least three isolate vertices in , which are target vertices in . is corresponding with nontarget vertices.(1)By Lemma 1, if , then the number of in a simple graph with vertices and edges reaches the maximum if the graph is either or . Thus, is either or , which means that is either or , where . So, we need to compare and , which can be calculated as the same analysis of . For convenient, if , then let be the graph . Then, and . By calculation, , and . So, Therefore, if , the graph is the unique locally most reliable graph in for close to 0.(2)By Lemma 1, if , then the number of in a simple graph with vertices and edges is maximized only if the graph is . Thus, is , which means that is . Therefore, if , then the graph is the unique locally most reliable graph in for close to 0.

Theorem 2. *Let , , and be positive integers. Then, is the unique locally most reliable graph in for close to 1.*

*Proof. *Let be the most reliable graph for close to 1. By Lemma 2, must have the largest edge connectivity of , and , which is the size of the smallest -cutset of as large as possible. For convenient, let be the set of three-terminal graph with the largest edge connectivity of , and . Obviously, .

For any graph , let be the minimal -cutset of , then there must exist a component containing just one target vertex and nontarget vertices in , without loss of generality, setting this target vertex as . Clearly, . If , then . Hence, can arrive at the maximum value if and only if . Thus, three target vertices in are adjacent to every vertex in . However, the connections among nontarget vertices are various. In order to get the precise construction of , by Lemma 2, we need to consider the coefficient . Obviously, , where is the number of the -cutsets of size . For given , is a constant and is corresponding with . Let be the minimum degree of . Since and , we have .

Continue to calculate the minimal -cutset of , , where is the edge number among nontarget vertices . If , then . If , then , where the equation holds if the components containing in are following two cases: one is composed of target vertex and one nontarget vertex with degree ; the other is composed of target vertex and nontarget vertices with degree and the induced subgraph by vertices in this component is . So, the number of with size is 0. Meanwhile, the number of with size is 3. By calculation, we see that for and , where is the number of nontarget vertices with degree and is the number of in the induced subgraph of nontarget vertices with degree .

Similarly, as the calculation of , we have for and , where is the number of nontarget vertices with degree and is the number of in the induced subgraph of nontarget vertices with degree in . Obviously, , and if , then for , and but . So, for and , which contradicts the assumption that is the most reliable graph for close to 1. Then, we have .

Clearly, . Since , the minimum degree of graph is larger than , which implies that for each , .

Therefore, is the unique locally most reliable graph in for close to 1.

As a straightforward consequence of Theorems 1 and 2, we obtain the following theorem.

Theorem 3. *Let and be positive integers. If , then there is no uniformly most reliable three-terminal graph in .*

Three specific classes of graphs with order and size are uniformly most reliable graphs.

*Remark 1. *For *n* = 4 and *m* = 4, there is a uniformly most reliable three-terminal graph in _{4,4} (see Example 1). For *n* = 5 and *m* = 8, there is a uniformly most reliable three-terminal graph is _{1} in _{5,8} (graphs are shown in Figure 6, and the comparison of the reliability polynomial is shown in Table 1). For n = 6 and m = 13, there is a uniformly most reliable three-terminal graph is _{1} in _{6,13} (graphs are shown in Figure 7, and the comparison of the reliability polynomial is shown in Table 2).

**(a)**

**(b)**

**(c)**

**(d)**

**(e)**

**(f)**

**(g)**

**(h)**

**(a)**

**(b)**

**(c)**

**(d)**

**(e)**

**(f)**

**(g)**

**(h)**

**(i)**

**(j)**

#### 4. The Uniformly Most Reliable Three-Terminal Graph

For the three-terminal graph with edges, it is complete graph, which is also the uniformly most reliable graph. In this section, we determine the uniformly most reliable graph in with .

When we remove one edge from with three target vertices, there are only three cases: the edge between target vertices; the edge between a target vertex and a nontarget vertex; and the edge between nontarget vertices. Let and be positive integers.(1)Using denotes the three-terminal graph on vertices and edges with vertex set and edge set .(2)Using denotes the three-terminal graph on vertices and edges with vertex set and edge set .(3)Using denotes the three-terminal graph on vertices and edges with vertex set and edge set .

Theorem 4. *Let and be positive integers. Then, is the unique uniformly most reliable graph in .*

*Proof. *By the definition of three-terminal reliability polynomial, if we can prove that there are more -subgraphs with edges in than in and for each , then is the unique uniformly most reliable graph in .

We complete this proof by constructing two injective maps and , from the -subgraphs with edges in and to the -subgraphs with edges in , respectively.

Construct the map :

Let be an -subgraph with edges in , where . According to whether , there are two cases we need to consider. Case 1: if does not contain the edge , then . The image is an -subgraph of with the same number of edges as . And this image does not contain the edge . Case 2: assume that contains the edge . Case 2.1: if is still an -subgraph, then . The image is an -subgraph of with the same number of edges as . Since this image contains the edge , it is distinct from Case 1. And is still an -subgraph. Case 2.2: if is not an -subgraph, but an -subgraph or an -subgraph, then . The image is an -subgraph of with the same number of edges as . Since the image contains and is not an -subgraph, it is distinct from the above cases. It is clear to see that in Case 2.2, contains either the edge and an edge for some , or the edge and an edge for some , or an edge and an edge for some . Case 2.3: assume that is not an -subgraph, or -subgraph, or -subgraph. In this case, all -subgraphs, -subgraphs, and -subgraphs in contain the edge . Thus, the image of the map defined by the above cases is not an -subgraph of . Let be a minimal -subgraph in . In , there are two components containing and , respectively. Targets and are in the same component; otherwise, is either an -subgraph or an -subgraph. Without loss of generality, let be in the component containing in and let be in the same component as in . Case 2.3.1: if is consisted of an edge , a minimal -subgraph, the edge , the edge , and a minimal -subgraph, then . does not have both edges and ; otherwise, an edge , an edge , and a -subgraph will get a -subgraph that does not contain the edge which contradicts the condition of Case 2.3. Therefore, has the same number of edges as . In , we have an -subgraph of which is consisted of an edge , a -subgraph, the edge , a -subgraph, and the edge . Since contains the edge but does not contain any edge and is not an -subgraph, is distinct from the above cases. The image in this case contains the edge . Case 2.3.2: if is consisted of an edge , a minimal -subgraph, the edge , and a minimal -subgraph, then . Similarly, as Case 2.3.1, does not have both edges and . Therefore, has the same number of edges as . In , we have an -subgraph of which is consisted of an edge , a -subgraph, a -subgraph, and the edge . Since contains edges and for some but does not contain any edge and is not an -subgraph, is distinct from the above cases. Case 2.3.3: if is consisted of the edge , the edge , the edge and a minimal -subgraph, then . Similarly, as Case 2.3.1, does not have both edges and . Therefore, has the same number of edges as . In , we have an -subgraph of is consisted of the edge , the edge , a -subgraph, and the edge . Since contains edges , , and but does not contain any edge and is not an -subgraph, is distinct from the above cases.Since the map defined on each of these cases yields -subgraph of as disjoint images, the map is injective.

Because there are at least as many -subgraphs with edges in as in for , is more reliable than for all .

Construct the map :

Let be an -subgraph with edges in , where . According to whether , there are two cases we need to consider. Case 1: if does not contain the edge , then . The image is an -subgraph of with the same number of edges as . And this image does not contain the edge . Case 2: assume that contains the edge . Case 2.1: if is still an -subgraph, then . The image is an -subgraph of with the same number of edges as . Since the image contains the edge , it is distinct from Case 1. And is still an -subgraph. Case 2.2: if is not an -subgraph but is an -subgraph and a -subgraph, or an -subgraph and an -subgraph, or an -subgraph and an -subgraph, then . The image is an -subgraph of