Edge-Neighbor-Rupture Degree of Graphs

Abstract

The edge-neighbor-rupture degree of a connected graph 𝐺 is defined to be ENR(𝐺)=max{𝜔(𝐺−𝑆)−|𝑆|−𝑚(𝐺−𝑆)∶𝑆⊆𝐸(𝐺),𝜔(𝐺−𝑆)≥1}, where 𝑆 is any edge-cut-strategy of 𝐺, 𝜔(𝐺−𝑆) is the number of the components of 𝐺−𝑆, and 𝑚(𝐺−𝑆) is the maximum order of the components of 𝐺−𝑆. In this paper, the edge-neighbor-rupture degree of some graphs is obtained and the relations between edge-neighbor-rupture degree and other parameters are determined.

1. Introduction

In a communication network, the vulnerability measures the resistance of the network to disruption of operation after the failure of certain stations or communication links. To measure the vulnerability we have some parameters which are connectivity [1], integrity [2], scattering number [3], and rupture degree [4].

A spy network can be modeled by a graph whose vertices represent the stations and whose edges represent the lines of communication. If a station is destroyed, the adjacent stations will be betrayed so that the betrayed stations become useless to network as a whole [5]. Therefore, instead of considering the stability of a communication network in standard sense, some new graph parameters such as vertex-neighbor-connectivity [6] and edge-neighbor-connectivity [7], vertex-neighbor-integrity [8] and edge-neighbor-integrity [9], vertex-neighbor-scattering number [10] and edge-neighbor-scattering number [11], and vertex-neighbor-rupture degree [12] were introduced to measure the stability of communication networks in “neighbor” sense.

We use Bondy and Murty [1] for terminology and notation not defined here and consider only finite simple connected graphs. Let 𝐺=(𝑉,𝐸) be a graph and 𝑒 any edge in 𝐺. The diameter of  𝐺, denoted by diam(𝐺), is the maximum distance over all pairs of vertices in 𝐺.

𝑁(𝑒)={𝑓∈𝐸(𝐺)∣𝑓≠𝑒;𝑒 and 𝑓 are adjacent} is the open-edge-neighborhood of 𝑒, and 𝑁[𝑒]=𝑁(𝑒)∪{𝑒} is the closed-edge-neighborhood of 𝑒. An edge 𝑒 in 𝐺 is said to be subverted when 𝑁[𝑒] is deleted from 𝐺. In other words, if 𝑒=[𝑢,𝑣], 𝐺−𝑁[𝑒]=𝐺−{𝑢,𝑣}. A set of edges 𝑆 is called an edge subversion strategy of 𝐺 if each of the edges in 𝑆 has been subverted from 𝐺. The survival subgraph is denoted by 𝐺−𝑆. An edge subversion strategy 𝑆 is called an edge-cut-strategy of 𝐺 if the survival subgraph 𝐺−𝑆 is disconnected or is a single vertex or the empty graph [13].

The edge-neighbor-connectivity of 𝐺, Λ(𝐺), is the minimum size of all edge-cut-strategies of 𝐺. A graph 𝐺 is m-edge-neighbor-connected if Λ(𝐺)=𝑚 [7].

The edge-neighbor-integrity of a graph 𝐺, ENI(𝐺), is defined to be ENI(𝐺)=min_{𝑆⊆𝐸(𝐺)}{|𝑆|+𝑚(𝐺−𝑆)}, (1) where 𝑆 is any edge subversion strategy of 𝐺 and 𝑚(𝐺−𝑆) is maximum order of the components of 𝐺−𝑆 [9].

The edge-neighbor-scattering number of 𝐺, ENS(𝐺), is defined as ENS(𝐺)=max_{𝑆⊆𝐸(𝐺)}{𝜔(𝐺−𝑆)−|𝑆|}, (2) where 𝑆 is any edge-cut-strategy of 𝐺 and 𝜔(𝐺−𝑆) is the number of the components of 𝐺−𝑆 [11].

The known parameters concerning the neighborhoods do not deal with the number of the removing edges, the number of the components, and the number of the vertices in the largest component of the remaining graph in a disrupted network simultaneously. In order to fill this void in the literature, the current study proposes a definition of edge-neighbor-rupture degree which is a new parameter concerning these three values. Additionally, this study also analyzes the relations between edge-neighbor-rupture degree and some other parameters and obtains edge-neighbor-rupture degree of some graphs.

The edge-neighbor-rupture degree of a connected graph 𝐺 is defined to be (𝐺){𝜔()−|𝑆|()(𝐺),𝜔()},ENR=max𝐺−𝑆−𝑚𝐺−𝑆∶𝑆⊆𝐸𝐺−𝑆≥1 (3) where 𝑆 is any edge-cut-strategy of 𝐺, 𝜔(𝐺−𝑆) is the number of the components of 𝐺−𝑆, and 𝑚(𝐺−𝑆) is the maximum order of the components of 𝐺−𝑆. A set 𝑆^∗⊆𝐸(𝐺) is said to be the ENR-set of 𝐺 if ENR(𝐺)=𝜔(𝐺−𝑆^∗)−|𝑆^∗|−𝑚(𝐺−𝑆^∗).

The edge-neighbor-rupture degree differs from edge- neighbor-connectivity, edge-neighbor-integrity, and edge-neighbor-scattering number in showing the vulnerability of networks. For example, consider the graphs 𝐺₁ and 𝐺₂ in Figure 1.

(a)

(b)

Figure 1 

The graphs 𝐺1 and 𝐺2.

It can be easily seen that the edge-neighbor-connectivity, edge-neighbor-integrity, and edge-neighbor-scattering number of these graphs are equal: Λ𝐺₁=Λ𝐺₂𝐺=1,ENI₁=ENI𝐺₂𝐺=4,ENS₁=ENS𝐺₂=4. (4)

On the other hand, the edge-neighbor-rupture degrees of 𝐺₁ and 𝐺₂ are different: ENR𝐺₁𝐺=1,ENR₂=2. (5)

Hence, the edge-neighbor-rupture degree is a better parameter for distinguishing vulnerability of graphs 𝐺₁ and 𝐺₂.

2. Bounds for Edge-Neighbor-Rupture Degree

In this section some lower and upper bounds are given for the edge-neighbor-rupture degree of a graph using different graph parameters.

Theorem 1. Let 𝐺 be a connected graph of order 𝑛. Then, 𝐸𝑁𝑅(𝐺)≤𝑛−4. (6)

Proof. Let 𝑆 be an edge-cut-strategy of 𝐺 and |𝑆|=𝑟. If 𝑟≥1, then 𝜔(𝐺−𝑆)≤𝑛−2 and 𝑚(𝐺−𝑆)≥1. Therefore, 𝜔(𝐺−𝑆)−|𝑆|−𝑚(𝐺−𝑆)≤𝑛−2−1−1. (7)
Hence we have ENR(𝐺)≤𝑛−4. (8)
The proof is completed.

Theorem 2. Let 𝐺 be a connected graph of order 𝑛, and let 𝛼(𝐺), Λ(𝐺) be the independent number and edge-neighbor-connectivity of 𝐺, respectively. Then, 𝐸𝑁𝑅(𝐺)≤𝛼(𝐺)−Λ(𝐺)−1. (9)

Proof. Let 𝑆 be an edge-cut-strategy of 𝐺. For any 𝑆 of 𝐺, |𝑆|≥Λ(𝐺), 𝜔(𝐺−𝑆)≤𝛼(𝐺), and 𝑚(𝐺−𝑆)≥1. Hence we get ENR(𝐺)≤𝛼(𝐺)−Λ(𝐺)−1. (10)
The proof is completed.

Theorem 3. Let 𝐺 be a connected graph of order 𝑛≥3. If 𝐸𝑁𝑅(𝐺)=𝑛−4; then diam(𝐺)≤3.

Proof. Assume that diam(𝐺)≥4; then 𝐺 contains a path 𝑃₅. Thus for any edge 𝑒 in 𝐺, 𝜔(𝐺−𝑆)≤𝑛−2, 𝑚(𝐺−{𝑒})≥2, and for any two edges 𝑒₁ and 𝑒₂ in 𝐺, 𝜔(𝐺−𝑒)≤𝑛−2, 𝑚(𝐺−{𝑒_𝑙,𝑒₂})≥1. Therefore ENR(𝐺)≤𝑛−5, (11) a contradiction. Hence diam(𝐺)≤3.
The proof is completed.

Theorem 4. Let 𝐺 be a connected graph of order 𝑛 and 𝛼^′(𝐺) edge independence number of 𝐺. Then, 𝐸𝑁𝑅(𝐺)≤𝑛−3𝛼^′(𝐺)−1. (12)

Proof. Let 𝑆 be an edge-cut-strategy of 𝐺. If |𝑆|=𝛼^′(𝐺), then 𝐺−𝑆 contains 𝑛−2𝛼^′(𝐺) isolated vertices and 𝑚(𝐺−𝑆)≥1. From the definition of edge neighbor rupture degree we have ENR(𝐺)≤𝑛−3𝛼^′(𝐺)−1. (13)
The proof is completed.

3. Edge-Neighbor-Rupture Degree of Some Graphs

In this section, we consider the edge-neighbor-rupture degree of some graphs.

Theorem 5. Let 𝑃_𝑛 be a path with order 𝑛(≥4). Then 𝐸𝑁𝑅𝑃_𝑛=(),().0,𝑛≡1mod3−1,𝑛≡0,2mod3 (14)

Proof. Let 𝑆 be an edge-cut-strategy of 𝑃_𝑛 and |𝑆|=𝑟. If 𝑟≤⌊(𝑛−1)/3⌋, then 𝜔(𝑃_𝑛−𝑆)≤𝑟+1 and 𝑚(𝑃_𝑛−𝑆)≥⌈(𝑛−2𝑟)/(𝑟+1)⌉. Thus 𝜔𝑃_𝑛−𝑆−|𝑆|−𝑚𝑃_𝑛−𝑆≤𝑟+1−𝑟−𝑛−2𝑟𝑃𝑟+1ENR_𝑛≤max_𝑟1−𝑛−2𝑟,𝑟+1 (15) the function 𝑓(𝑟) takes its maximum value at 𝑟=⌊(𝑛−1)/3⌋, and we get ENR(𝑃_𝑛)≤−1 where 𝑛≡0,2(Mod3) and ENR(𝑃_𝑛)≤0 where 𝑛≡1(Mod3). So, we have ENR𝑃_𝑛≤(),().0,𝑛≡1mod3−1,𝑛≡0,2mod3 (16)
On the other hand, if 𝑟≥⌊(𝑛−1)/3⌋+1, then we have 𝜔(𝑃_𝑛−𝑆)≤𝑟 and 𝑚(𝑃_𝑛−𝑆)≥1. Hence 𝜔𝑃_𝑛−𝑆−|𝑆|−𝑚𝑃_𝑛𝑃−𝑆≤𝑟−𝑟−1,ENR_𝑛≤−1. (17)
It can be easily seen that there is an edge-cut-strategy 𝑆^∗ of 𝑃_𝑛 such that |𝑆^∗|=⌊(𝑛−1)/3⌋, 𝜔(𝑃_𝑛−𝑆^∗)=⌊(𝑛−1)/3⌋+1, and 𝑚(𝑃_𝑛−𝑆^∗)=2 where 𝑛≡0,2(Mod3) and 𝑚(𝑃_𝑛−𝑆^∗)=1 where 𝑛≡1(Mod3). Therefore, ENR𝑃_𝑛=(),().0,𝑛≡1mod3−1,𝑛≡0,2mod3 (18)
The proof is completed by (16), (17), and (18).

Theorem 6. Let 𝐶_𝑛 be a cycle with order 𝑛(≥6). Then 𝐸𝑁𝑅𝐶_𝑛=(),().−1,𝑛≡0mod3−2,𝑛≡1,2mod3 (19)

Proof. Let 𝑆 be an edge-cut-strategy of 𝐶_𝑛 and |𝑆|=𝑟. If 𝑟≤⌊𝑛/3⌋, then 𝜔(𝐶_𝑛−𝑆)≤𝑟 and 𝑚(𝐶_𝑛−𝑆)≥⌈(𝑛−2𝑟)/𝑟⌉. Thus 𝜔𝐶_𝑛−𝑆−|𝑆|−𝑚𝐶_𝑛−𝑆≤𝑟−𝑟−𝑛−2𝑟𝑟,ENR𝐶_𝑛≤max_𝑟𝑛−2𝑟𝑟, (20) the function 𝑓(𝑟) takes its maximum value at 𝑟=⌊𝑛/3⌋, and we get ENR(𝐶_𝑛)≤−1 where 𝑛≡0(Mod3) and ENR(𝐶_𝑛)≤−2 where 𝑛≡1,2(Mod3). So, we have ENR𝐶_𝑛≤(),().−1,𝑛≡0mod3−2,𝑛≡1,2mod3 (21)
On the other hand, if 𝑟≥⌊𝑛/3⌋+1, then we have 𝜔(𝐶_𝑛−𝑆)≤𝑟−1 and 𝑚(𝐶_𝑛−𝑆)≥1. Hence 𝜔𝐶_𝑛−𝑆−|𝑆|−𝑚𝐶_𝑛𝐶−𝑆≤𝑟−1−𝑟−1ENR_𝑛≤−2. (22)
It can be easily seen that there is an edge-cut-strategy 𝑆^∗ of 𝐶_𝑛 such that |𝑆^∗|=⌊𝑛/3⌋,𝜔(𝐶_𝑛−𝑆^∗)=⌊𝑛/3⌋, and 𝑚(𝐶_𝑛−𝑆^∗)=2 where 𝑛≡1,2(Mod3) and 𝑚(𝐶_𝑛−𝑆^∗)=1 where 𝑛≡0(Mod3). Therefore ENR𝐶_𝑛=(),().−1,𝑛≡0mod3−2,𝑛≡1,2mod3 (23)
The proof is completed by (21), (22), and (23).

Lemma 7 (see [7]). For any graph 𝐺 with order 𝑛, Λ(𝐺)≤⌊𝑛/2⌋.

Theorem 8. Let 𝐾_𝑛 be a complete graph with order 𝑛. Then 𝐸𝑁𝑅𝐾_𝑛𝑛=−2. (24)

Proof. Let 𝑆 be an edge-cut-strategy of 𝐾_𝑛 and |𝑆|=𝑟. By Lemma 7 we know Λ(𝐾_𝑛)=⌊𝑛/2⌋. If 𝑟≥⌊𝑛/2⌋, then 𝜔(𝐾_𝑛−𝑆)≤1 and 𝑚(𝐾_𝑛−𝑆)≥1. Hence, 𝜔𝐾_𝑛−𝑆−|𝑆|−𝑚𝐾_𝑛𝑛−𝑆≤1−2𝐾−1,ENR_𝑛𝑛≤−2. (25)
It can be easily seen that there is an edge set 𝑆^∗ of 𝐾_𝑛 such that |𝑆^∗|=⌊𝑛/2⌋, then we have 𝜔(𝐾_𝑛−𝑆)=1 and 𝑚(𝐾_𝑛−𝑆)=1. From the definition of edge-neighbor-rupture degree we have ENR𝐾_𝑛𝑛=−2. (26)
The proof is completed.

Definition 9. We also call 𝐾_1,𝑛 a star with 𝑛+1 vertices. Let DS(𝑛_𝑙,𝑛₂) be a double star with {𝑛_𝑙,𝑛₂} end-vertices, where 𝑛₁≥0 and 𝑛₂≥0, and a common edge [𝑢,𝑣], as shown in Figure 2. Note that if either 𝑛₁ or 𝑛₂ is 0, then the double star DS(𝑛_𝑙,𝑛₂) is a star.

Figure 2 

DS(𝑛𝑙,𝑛2).

Theorem 10. Let 𝑇 be a tree of order 𝑛≥4. Then 𝐸𝑁𝑅(𝑇)=𝑛−4 if and only if 𝑇 is either a star 𝐾_1,𝑛−1 or a double star DS(𝑛_𝑙,𝑛₂), where 𝑛₁≥1, 𝑛₂≥1, and 𝑛₁+𝑛₂=𝑛−2.

Proof. If 𝑇 is a tree of order 𝑛≥3 and ENR(𝑇)=𝑛−4, then by Theorem 3 we have either diam(𝑇)=2 or diam(𝑇)=3. If diam(𝑇)=2, then  𝑇 is a star 𝐾_1,𝑛−1. If diam(𝑇)=3, then 𝑇 is a double star DS(𝑛_𝑙,𝑛₂), where 𝑛₁>0, 𝑛₂>0, and 𝑛₁+𝑛₂=𝑛−2.
Conversely, let 𝑇 be either a star 𝐾_1,𝑛−1 with the order 𝑛≥4 or a double star DS(𝑛_𝑙,𝑛₂), where 𝑛₁≥1, 𝑛₂≥1, and 𝑛₁+𝑛₂+2=𝑛≥4.
If 𝑆 is an edge-cut-strategy of 𝐾_1,𝑛−1 and |𝑆|=1, then we have 𝑛−2 isolated vertices. Therefore we have 𝜔(𝑇−𝑆)=𝑛−2 and 𝑚(𝑇−𝑆)=1. So 𝜔𝐾_1,𝑛−1−𝑆−|𝑆|−𝑚𝐾_1,𝑛−1𝐾−𝑆=𝑛−2−1−1ENR_1,𝑛−1=𝑛−4. (27)
It can be easily seen that there is an edge-cut-strategy 𝑆^∗ of DS(𝑛_𝑙,𝑛₂) such that |𝑆^∗|=1; then we have 𝜔(DS(𝑛_𝑙,𝑛₂)−𝑆^∗)=𝑛−2 and 𝑚(DS(𝑛_𝑙,𝑛₂)−𝑆^∗)=1. So 𝜔DS𝑛_𝑙,𝑛₂−𝑆^∗−𝑆^∗−𝑚DS𝑛_𝑙,𝑛₂−𝑆^∗𝑛=𝑛−2−1−1,ENRDS_𝑙,𝑛₂=𝑛−4. (28)
If 𝑆 is an edge set of DS(𝑛_𝑙,𝑛₂) and |𝑆|>1, then we have 𝜔(DS(𝑛_𝑙,𝑛₂)−𝑆)=1 and 𝑚(DS(𝑛_𝑙,𝑛₂)−𝑆)≥2.
The proof is completed.

Theorem 11. Let 𝐾_𝑚,𝑛 be a complete bipartite graph with |𝑚−𝑛|≥1. Then 𝐸𝑁𝑅𝐾_𝑚,𝑛=𝑚−2𝑛−1,𝑖𝑓𝑚>𝑛,𝑛−2𝑚−1,𝑖𝑓𝑚<𝑛. (29)

Proof. Assume 𝑚>𝑛. Let 𝑆 be an edge-cut-strategy of 𝐾_𝑚,𝑛 and |𝑆|=𝑟. If 𝑟≥𝑛, then 𝜔(𝐾_𝑚,𝑛−𝑆)≤𝑚−𝑛 and 𝑚(𝐾_𝑚,𝑛−𝑆)≥1. Hence, 𝜔𝐾_𝑚,𝑛−𝑆−|𝑆|−𝑚𝐾_𝑚,𝑛−𝑆≤𝑚−𝑛−𝑟−1, (30) the function 𝑓(𝑟) is a decreasing function and takes its maximum value at 𝑟=𝑛, and we get ENR𝐾_𝑚,𝑛≤𝑚−2𝑛−1. (31)
It can be easily seen that there is an edge set 𝑆^∗ of 𝐾_𝑚,𝑛 such that |𝑆^∗|=𝑛; then we have 𝜔(𝐾_𝑚,𝑛−𝑆)=𝑚−𝑛 and 𝑚(𝐾_𝑚,𝑛−𝑆)=1. From the definition of edge neighbor rupture degree we have ENR𝐾_𝑚,𝑛=𝑚−2𝑛−1. (32)
Similarly, we obtain ENR(𝐾_𝑚,𝑛)=𝑛−2𝑚−1 when 𝑛>𝑚.
Finally, we have ENR𝐾_𝑚,𝑛=𝑚−2𝑛−1,if𝑚>𝑛,𝑛−2𝑚−1,if𝑚<𝑛. (33)
The proof is completed.

Definition 12. The wheel graph with 𝑛 spokes, 𝑊_𝑛, is the graph that consists of an n-cycle and one additional vertex, say 𝑢, that is adjacent to all the vertices of the cycle. In Figure 3, we display 𝑊₆.

Figure 3 

The wheel graph 𝑊6.

Theorem 13. Let 𝑊_𝑛 be a wheel graph with order 𝑛(≥5). Then 𝐸𝑁𝑅𝑊_𝑛=(),().−1,𝑛≡2mod3−2,𝑛≡0,1mod3 (34)

Proof. The graph 𝑊_𝑛 has subgraphs 𝐶_𝑛 and 𝐾_1,𝑛. Let 𝑒 be any one edge of 𝐾_1,𝑛.
If 𝑒∈𝑆 and |𝑆|=1, then we get 𝜔(𝑊_𝑛−𝑆)=𝑃_𝑛−1. So,ENR𝑊_𝑛=ENR𝑃_𝑛−1−1. (35)
If 𝑒∉𝑆 and |𝑆|=1, then 𝜔(𝑊_𝑛−𝑆)=1 and 𝑚(𝑊_𝑛−𝑆)≥2. Hence, ENR𝑊_𝑛≤𝜔𝑊_𝑛−𝑆−|𝑆|−𝑚𝑊_𝑛−𝑆≤1−1−2=−2. (36)
If 𝑒∉𝑆 and |𝑆|≥2, then 𝜔(𝑊_𝑛−𝑆)=1 and 𝑚(𝑊_𝑛−𝑆)≥1. Thus, ENR𝑊_𝑛≤𝜔𝑊_𝑛−𝑆−|𝑆|−𝑚𝑊_𝑛−𝑆≤1−2−1=−2. (37)
The proof is completed by (35), (36), and (37).