Uniformly most reliable three-terminal graph of dense graphs

Suppose that every edge of a graph $G$ survives independently with a fixed probability between $0$ and $1$. The three-terminal reliability is the connection probability of the fixed three target vertices $r,s$ and $t$ in a three-terminal graph. This research focuses on the uniformly most reliable three-terminal graph of dense graphs with $n$ vertices and $m$ edges in some ranges. First, we give the locally most reliable three-terminal graphs of $n$ and $m$ in a certain range for $p$ close to $0$ and for $p$ close to $1$. And then, we prove that there is no uniformly most reliable three-terminal graph with certain ranges of $n$ and $m$. Finally, some uniformly most reliable graphs are given for $\binom{n}{2}-2$ ($4\leq n\leq 6$) and $\binom{n}{2}-1$ ($n\geq5$). This study of uniformly or locally most reliable three-terminal graph provides helpful guidance for constructing highly reliable network structures involving three key vertices as target vertices.


Introduction
In many applications, the reliability aspect of a network with n vertices and m edges can be modeled as a graph G with the same number of vertices, edges, and interconnections as the network. For all-terminal reliability (connection probability of all vertices of a graph), many studies have been done to determine the existence of a uniformly most reliable (all-terminal) graph for various values of n and m [1,3,4,7,8,13,14,16]. However, the research on k-terminal reliability (connection probability of a given set of target vertices with size k in a graph) is mainly about the algorithm of computing the k-terminal reliability polynomial [5,6,9,10,12], and only a few results on the construction of the most reliable k-terminal graph.
For large m (m is the number of edges of a graph), it is clear that there is only one graph for a complete graph (that is, m = n 2 ) and a complete graph with one edge removed (that is, m = n 2 − 1), so they are the uniformly most reliable graphs. In [16], it is shown that for n 2 − ⌊ n 2 ⌋ ≤ m ≤ n 2 − 2 (⌊ n 2 ⌋ is the largest integer not greater than n 2 ), the uniformly most reliable graph is a complete graph with a matching removed (the matching of a graph is a set of edges in a graph that have no common vertices with each other). These results have great significance in network design practices. In fact, in a real network, the design of the network often only needs to ensure the connectivity of k (2 ≤ k < n) key vertices (target vertices) in the network. Therefore, the construction of the most reliable k-terminal graph has high application value. However, by the literature available to the authors of the present research, there is a few research work on the construction of the most reliable k-terminal structure.
Betrand et al. [2] demonstrated in 2018 that for n 2 − ⌊ n−2 2 ⌋ ≤ m ≤ n 2 − 2, there is no uniformly most reliable two-terminal graph and for m = n 2 − 1, the uniformly most reliable two-terminal graph is a complete graph with an edge between non-target vertices removed.
Therefore, it is natural to consider the following problem.
Problem. For large m, is there a uniformly most reliable three-terminal graph? If it exists, how is it constructed? If it does not exist, can we construct the locally most reliable three-terminal graph and how to construct it?
With these questions, we further study the existence of uniformly most reliable threeterminal graphs for large m. For three-terminal graphs with m in the range n 2 − ⌊ n−3 2 ⌋ ≤ m ≤ n 2 − 2 (n ≥ 7), it is proved that there is no uniformly most reliable graph, and the locally most reliable three-terminal graphs are determined, one case is for p close to 0 and the other is for p close to 1. It also determines the uniformly most reliable three-terminal graph with n 2 − 2 (4 ≤ n ≤ 6) and n 2 − 1 (n ≥ 5) edges, respectively.
This present research is organized as follows. In section 2, some related basic definitions and notations are given. In section 3, the locally most reliable three-terminal graphs for m in a certain range are determined and show that there is no uniformly most reliable threeterminal graph for n 2 − ⌊ n−3 2 ⌋ ≤ m ≤ n 2 − 2 (n ≥ 7) and give the uniformly most reliable graphs for n = 4, 5, 6 and m = n 2 − 2. In section 4, a uniformly most reliable three-terminal graph with n vertices and m = n 2 − 1 edges is determined. Section 5 summarizes the results of this research.

Basic concepts and notations
A graph G = (V (G), E(G)) with three specified target vertices r, s and t in V (G) is a threeterminal graph. Using G n,m denotes the set of all simple three-terminal graphs with n vertices and m edges. The connectivity probability of the three specified target vertices r, s, t in graph G ∈ G n,m when each edge of G survives independently with a fixed probability p is called the three-terminal reliability of G, or the three-terminal reliability polynomial of G, denote by R 3 (G; p). A v 1 v 2 · · · v n -subgraph is a subgraph of G in which vertices v 1 v 2 · · · v n are connected in the subgraph. The three-terminal reliability polynomial of the graph G ∈ G n,m can be written as where N i (G) or simply N i is the number of rst-subgraphs of graph G with i edges.
Example 1. Figure 1 shows all types of simple three-terminal graph in G 4,4 with three target vertices r, s, t. Each edge of these graphs survives independently with probability p. Thus N 2 (G 1 ) = 3, N 3 (G 1 ) = 4 and N 4 (G 1 ) = 1.
Similarly, by definition, we can calculate N i (G j ), 2 ≤ i, j ≤ 4, are as follows: Figure 2 shows a visualization among all graphs in G 4,4 . Clearly, for all 0 < p < 1, Example 2. Figure 3 shows two special simple three-terminal graphs in G 8,26 with three target vertices r, s, t. Each edge of these graphs survives independently with probability p.
In fact, this research later proved that H 1 is the locally most reliable graph for p close to 1 and H 2 is the locally most reliable graph for p close to 0 in G 8,26 .  In fact, many researches on reliability focuses on determining a uniformly most reliable graph for a given number of vertices n and edges m, as shown in Example 1; if there is no uniformly most reliable graph, researchers usually focuses on determining the locally most reliable graph for p close to 0 or 1, as shown in Example 2. So, similar to the definition of uniformly (locally) most reliable graph [1,4], we give the following definition of uniformly (locally) most reliable three-terminal graph.
for p close to 0 (for p close to 1), then G is the locally most reliable graph in G n,m for p close to 0 (for p close to 1) Here are some notations used in the following. If G is a simple graph, let N G (H) denote the number of subgraphs as H, whose vertices is the subset of non-target vertices in G, and In addition, P n is the path with n vertices, and let K n denote the complete graph with n vertices, in which there is exactly one edge between each pair of vertices, and K 1,n denotes a star with n + 1 vertices and n edges.

Some locally most reliable three-terminal graphs
In this section, the locally most reliable three-terminal graph for n 2 − (n − 4) ≤ m ≤ n 2 − 2 for p close to 0 is determined and the locally most reliable three-terminal graph for n 2 − ⌊ n−3 2 ⌋ ≤ m ≤ n 2 − 2 for p close to 1 is also determined. Then, it is shown that for n ≥ 7 and n 2 − ⌊ n−3 2 ⌋ ≤ m ≤ n 2 − 2, there is no uniformly most reliable graph in G n,m , and for n = 4, 5, 6 and m = n 2 − 2, there is a uniformly most reliable graph. To prove these results, we first introduce some related definitions and lemmas.
If the rst-subgraph with i edges does not contain any rst-subgraph with less than i edges, then it is minimal, otherwise it is non-minimal. A rst-cutset is a set of edges whose deletion results in the disconnection of vertices r, s and t in the graph and the number of edges is its size. The edge connectivity of r, s and t is the smallest size of a rst-cutset, denoted by λ(rst) or simply λ.
In general, the calculation of the three-terminal reliability polynomial of a graph is NPhard [15,17]. Therefore, we study the locally most reliable graph by the following lemma.
Let N i (G) = N i (H) for 1 ≤ i < k and for l < i ≤ m and k ≤ l, where k and l are integers.
Proof. Assume that G and H satisfy the given conditions, we have By Lemma 3.1, we get the following conclusions.
(1) If G ∈ G n,m is the locally most reliable graph for p close to 0, then it must contain the triangle rst and the value of N 3 (G) is the maximum among all graphs in G n,m containing the triangle rst.
(2) If G ∈ G n,m is the locally most reliable graph for p close to 1, then it must have the largest edge connectivity λ.
and N m−λ = m λ − the number of the rst-cutsets of size λ, the number of the rst-cutsets with size λ of graph G must be minimized. Now, we demonstrate the locally most reliable graph for three-terminal graphs. We first introduce two important graphs for this section, as follows.
Let n ≥ 7 and 2 ≤ l ≤ n − 4 be positive integers. Using A n,l denotes the three-terminal graph on n vertices and n Let n ≥ 7 and 2 ≤ l ≤ ⌊ n−3 2 ⌋ be positive integers. Using A ′ n,l denotes the three-terminal graph on n vertices and n 2 − l edges with vertex set Figure 5 depicts these two three-terminal graphs with 11 vertices and 51 edges.
If m = 3, there are two simple graphs with the maximum number of P 3 : K 3 ∪ K n−3 and We give the following Theorem 3.1. Proof. Suppose that n, l and m satisfy the conditions and G is the locally most reliable graph in G n,m for p close to 0 and the vertex set is Then by Lemma 3.1, G must contain the triangle rst and the value of N 3 must reach the maximum among all graphs in G n,m containing the triangle rst.
It is no hard to see that Clearly, for all graphs in G n,m containing the triangle rst, a = 3 and b = 3(m − 3) are constants and N 3 take the maximum value if and only if c and d attains the maximum value.
Note that if d takes the maximum value n − 3, then the value of c also reaches its maximum, Now, consider the remaining n 2 − l − 3n − 6 edges between non-target vertices in G that have not been described. Since G is a dense graph, it is often easier to consider the position of the l edges deleted between non-target vertices.
By Lemma 3.1, we need to continue to calculate the coefficients N i = b i + c i , where b i and c i are the number of minimal rst-subgraphs and non-minimal rst-subgraphs with i edges, respectively. We now calculate  Note that the value of n i=4 d i 2 is the number of subgraphs as P 3 , whose vertices is the subset of non-target vertices in G.
(1) By Lemma 3.2, if l = 3, then the number of P 3 in a simple graph with n − 3 vertices and l edges reaches the maximum if the graph is either K 3 ∪ K n−6 or K 1,3 ∪ K n−7 . So, we get that N 5 (A n,3 ) = N 5 (A * n,3 ) attains the maximum among the graphs whose N i (1 ≤ i ≤ 4) satisfy the above calculations, that is, G is either A n,3 or A * n,3 . So, we need to calculate the value of N 6 (A n,3 ) − N 6 (A * n,3 ).
Similar to the analysis and solution process of N 5 , it can be calculated, By calculation, we can get N A n,3 ( Therefore, if l = 3, then the graph A * n,l is the graph G which is the unique locally most reliable graph in G n,m for p close to 0. (2) By Lemma 3.2, if l = 3, then the number of P 3 in a simple graph with n − 3 vertices and l edges is maximized only if the graph is K 1,l ∪ K n−4−l . So, we get that N 5 (A n,l ) attains the maximum among the graphs whose N i (1 ≤ i ≤ 4) satisfy the above calculations.
Therefore, if l = 3, then the graph A n,l is the graph G which is the unique locally most reliable graph in G n,m for p close to 0. Now, we will show that for n 2 − ⌊ n−3 2 ⌋ ≤ m ≤ n 2 − 2 (n ≥ 7), A ′ n,l is the unique locally most reliable graph in G n,m for p close to 1, as shown in Theorem 3.2. To prove this, we need to give some lemmas first.
Proof. Suppose that G ∈ G n,m and C satisfy the given hypotheses. Then when all the components obtained by G − C are both complete graphs, the number of edges in the graph G − C is the maximum. There are two cases of components obtained by the graph G − C: Case 1. Obtain two components: one containing r (or rs or rt), and the other containing st (or t or s).

Case 2. Obtain three components containing r, s, t respectively.
In Case 1, the two components contain k + 1 and n − k − 1 vertices, respectively. Thus, the number of edges in G− C is k+1 2 + n−k−1

2
. In Case 2, since the component containing s has By calculation, Thus, the maximum number of the edges in G − C is k+1 On the other hand, the minimum number of the edges in G−C is k+1 Thus, we have The proof is thus complete.
Lemma 3.4 For positive integers n ≥ 7 and 2 ≤ l ≤ ⌊ n−3 2 ⌋ and every graph G ∈ G n,m with m = n 2 − l edges, the smallest rst-cutset of G contains all edges incident with one of the target vertices r, s, t and the next smallest minimal rst-cutset of G will obtain an order 2 components containing either r or s or t when it is removed.
Proof. By Lemma 3.3, if C is a minimal rst-cutset of G, then the component containing either r or s or t in G − C has k + 1 (0 ≤ k ≤ ⌊ n−3 2 ⌋) vertices. Without loss of generality, let C be a minimal rst-cutset of G and the component containing r in G − C has k + 1 It is easy to see that k(n − k − 2) increases as 0 ≤ k ≤ ⌊ n−3 2 ⌋ increases, so, we have If k = 0 (that is, r is the unique vertex in the component that contains r in G − C), then Since for k = 0, |C| ≤ n − 1 when k ′ = 0 and n − 1 < 3n−5 2 (n ≥ 7), so, when k = 0, the smallest rst-cutset of G can be obtained, that is, the smallest rst-cutset of G contains all the edges incident with either r or s or t.
If k = 1 (that is, there exist r and one other vertex in the component that contains r in If k ≥ 2, then Since for k = 1, |C| ≤ 2n − 4 when k ′ = 0 and 2n − 4 ≤ 5n−15 2 (n > 7) and no hard to get that the size of the next smallest minimal rst-cutset of G for k = 1 is smaller than for k ≥ 2 (n = 7), so, when k = 1, the next smallest minimal rst-cutset of G can be obtained, that is, the next smallest minimal rst-cutset of G will obtain an order 2 components containing either r or s or t when it is removed.
The proof is now complete.
Theorem 3.2 Let n ≥ 7, 2 ≤ l ≤ ⌊ n−3 2 ⌋ and m = n 2 − l be positive integers. Then A ′ n,l is the unique locally most reliable graph in G n,m for p close to 1.
Proof. Assume that n, l and m satisfy the given hypotheses. Let G ∈ G n,m be the unique most reliable graph for p close to 1. Then by Lemma 3.1, G must have the largest edge connectivity λ, that is, the size of the smallest rst-cutset of G must be as large as possible.
By Lemma 3.4, we get that for G, λ = d(r) = d(s) = d(t) = n − 1. There are many graphs satisfying this condition, so, the size of the next smallest minimal rst-cutset of G also must be as large as possible with λ = n − 1.
By Lemma 3.4, the next smallest minimal rst-cutset leaves r and one other vertex v in a component, whose size is Therefore, A ′ n,l is the unique locally most reliable graph in G n,m for p close to 1.
As a straightforward consequence of Theorems 3.1 and 3.2, we obtain the following Theorem 3.3.

A uniformly most reliable three-terminal graph
For the three-terminal graph with m = n 2 edges, there is only one graph, thus, it is easy to see that it is the uniformly most reliable graph in G n,( n 2 ) . In this section, we determine a uniformly most reliable graph in G n,m with m = n 2 − 1 edges. First, we introduce three graphs used in the following Theorem.
Clearly, when we remove one edge, there are only three distinct cases: the edge between target vertices; the edge between a target vertex and a non-target vertex; the edge between non-target vertices. Let n ≥ 5 and m = n 2 − 1 be positive integers.
(1) Using X n denotes the three-terminal graph on n vertices and m edges with vertex (2) Using Y n denotes the three-terminal graph on n vertices and m edges with vertex set V (Y n ) = {r = y 1 , s = y 2 , t = y 3 , y 4 , · · · , y n } and edge set E(Y n ) = {y i y j |1 ≤ i < j ≤ n} − {ry 4 }.
(3) Using Z n denotes the three-terminal graph on n vertices and m edges with vertex Now, we can give a uniformly most reliable graph in G n,m for n ≥ 5 and m = n 2 − 1, as shown in Theorem 4.1.

Proof.
To prove this theorem, we will prove that there are more rst-subgraphs with i (2 ≤ i ≤ n 2 − 1) edges in Z n than in X n and Y n .
We complete this proof by construct two injective maps f X and f Y , from the rst-subgraphs with i edges in X n and Y n to the rst-subgraphs with i edges in Z n , respectively.
Construct the map f X : Let S be a rst-subgraph with i edges in X n , where 2 ≤ i ≤ n 2 − 1. The image is a rst-subgraph of Z n with the same number of edges as S. And this image does not contain the edge rs.
Case 2. Assume that S contains the edge x 4 x 5 .
The image is a rst-subgraph of Z n with the same number of edges as S. Since this image contains the edge rs, it is distinct from Case 1. And f X (S) − {rs} is still a rst-subgraph.
is not a rst-subgraph, but a rt-subgraph or a st-subgraph, then The image is a rst-subgraph of Z n with the same number of edges as S. Since the image contains rs and f X (S) − {rs} is not a rst-subgraph, it is distinct from the above cases. It is clear to see that in f X (S), it contains either the edge st and an edge rz i for some 4 ≤ i ≤ n or the edge rt and an edge sz j for some 4 ≤ j ≤ n or an edge rz i and an edge sz j for some It is easy to see that for this case, all rst-subgraph and all rt-subgraph and all st-subgraph in S contains the edge x 4 x 5 . Thus, the image of the map defined by the above cases is not a rst-subgraph of Z n . Let S ′ be a minimal rst-subgraph in S. Then S ′ consists of a minimal rsx 4 -subgraph, the edge x 4 x 5 and a minimal x 5 t-subgraph.
Similarly, S does not have both edges x 5 x j and rx j . Therefore, f X (S) has the same size as S. In f X (S), we have a rst-subgraph of Z n which consists of an edge z 5 z j , a sz 4 z j -subgraph, a z 5 t-subgraph and the edge rs. Since f X (S) contains the edge rs and f X (S) − {rs} is not a rst-subgraph and it contains an edge sz j for some 4 ≤ j ≤ n and does not contain any edge , it is distinct from the above cases.
Case 2.3.3. If S ′ consists of the edge rx 4 , the edge sx 4 , the edge x 4 x 5 and a minimal It is easy to see that f X (S) has the same size as S. In f X (S), we have a rst-subgraph of Z n consists of the edge rz 4 , the edge rz 5 , a z 5 t-subgraph and the edge rs. Since the image has no inverse image in the above mappings, it is distinct from the above cases.
Therefore, all of these mappings are different. Since the map f X (S) defined on each of these cases of Z n as disjoint images, the map is injective.
Because there are at least as many rst-subgraphs with i edges in Z n as in X n for 2 ≤ i ≤ n 2 − 1, Z n is more reliable than X n for all p (0 ≤ p ≤ 1).
Construct the map f Y : Let S be a rst-subgraph with i edges in Y n , where 2 ≤ i ≤ n 2 − 1. The image is a rst-subgraph of Z n with the same number of edges as S. Since the image contains rz 4 and f Y (S) − {rz 4 } is not a rst-subgraph, it is distinct from the above cases.
Since S contains an edge ry j for some 2 ≤ j ≤ n and j = 4, the image also contains an edge rz j for some 2 ≤ j ≤ n and j = 4. Case 2.3. Assume that S − {y 4 y 5 } does not satisfy all of the following four cases: a rst-subgraph; a rsy 5 -subgraph and a y 4 t-subgraph; a ry 5 -subgraph and a sty 4 -subgraph; a rty 5 -subgraph and a sy 4 -subgraph.
It is easy to see that for this case, all rst-subgraph in S contains the edge y 4 y 5 . And S − {y 4 y 5 } is either a rsy 4 -subgraph and a y 5 t-subgraph, or a rty 4 -subgraph and a sy 5subgraph, or a ry 4 -subgraph and a sty 5 -subgraph. Therefore, the image of the map defined for the above cases is not a rst-subgraph of Z n . Let S ′ be a minimal rst-subgraph in S. According to the condition of S, S does not have both edges y 5 y j and ry j . Because if S contains both edges y 5 y j and ry j , then an edge y 5 y j , an edge ry j , a sy 4 y j -subgraph and a y 5 t-subgraph will get a rst-subgraph that does not contain the edge y 4 y 5 . Therefore, f Y (S) has the same size as S. In f Y (S), we have a rst-subgraph of Z n which consists of an edge z 5 z j , a sz 4 z j -subgraph, a z 5 t-subgraph and the edge rz 4 . Since f Y (S) contains the edge rz 4 and f Y (S) − {rz 4 } is not a rst-subgraph and it does not contain any edge rz j (2 ≤ j ≤ n, j = 4), it is distinct from the above cases. In f Y (S), any rs-subgraph does not contain z 5 . Similarly, S does not have both edges y 5 y j and ty j . Therefore, f Y (S) has the same size as S. In f Y (S), we have a rst-subgraph of Z n which consists of an edge z 5 z j , a tz 4 z j -subgraph, Similarly, S does not have both edges y 5 y j and ry j . Therefore, f Y (S) has the same size as S. In f Y (S), we have a rst-subgraph of Z n which consists of an edge z 5 z j , a z 4 z j -subgraph, a Therefore, all of these mappings are different. Since the map f Y (S) defined on each of these cases of Z n as disjoint images, the map is injective.
Because there are at least as many rst-subgraphs with i edges in Z n as in Y n for 2 ≤ i ≤ n 2 − 1, Z n is more reliable than Y n for all p (0 ≤ p ≤ 1).
From the above argument, we conclude that the graph Z n is the unique most reliable graph in G n,m for all p (0 ≤ p ≤ 1).

Conclusion
This research focuses on determining the existence of the uniformly most reliable graph for three-terminal graphs with number of edges in a given large range. If there is a uniformly most reliable graph, the uniformly most reliable graph is given; if there is no uniformly most reliable graph, the locally most reliable graphs are given. Based on the results of this research, the following conclusions can be drawn.
• When the number of vertices is n = 4 or 5 or 6 and the number of edges is m = n 2 −2, the uniformly most reliable graph is determined with comparisons in Example 1 and Appendix A.
• Under the conditions of n ≥ 7 and n 2 − (n − 4) ≤ m ≤ n 2 − 2, the locally most reliable graph in G n,m for p close to 0 is determined with proofs and for n 2 − ⌊ n−3 2 ⌋ ≤ m ≤ n 2 − 2, the locally most reliable graph in G n,m for p close to 1 is also determined with proofs. Then it shows that there is no uniformly most reliable three-terminal graph for n 2 − ⌊ n−3 2 ⌋ ≤ m ≤ n 2 − 2. It is worth considering whether there is a uniformly most reliable graph in the class of three-terminal graphs which delete more edges.
• With a complex proof, the uniformly most reliable graph in G n,( n 2 )−1 is determined, which is a graph with n 2 edges that removes an edge between non-target vertices. This conclusion is significant in comparison with the conclusion given by Betrand et al. [2], which states that the uniformly most reliable graph with n 2 − 1 edges for two-terminal graphs is also a graph with n 2 edges that removes an edge between non-target vertices. By these comparison, for m = n 2 − 1, it is most probably that the uniformly most reliable graph for k-terminal graphs is a graph with n 2 edges that removes an edge between non-target vertices.
The results of the research provide guiding significance for characterizing and determining the uniformly most reliable graphs or the locally most reliable graphs of general k-terminal networks. In fact, the results of the research can be useful for designing highly reliable networks with three key vertices (target vertices). r