Edge Neighbor Toughness of Graphs

Abstract

A new graph parameter, edge neighbor toughness is introduced to measure how difficult it is for a graph to be broken into many components through the deletion of the closed neighborhoods of a few edges. Let $G=(V,E)$ be a graph. An edge e is said to be subverted when its neighborhood and the two endvertices are deleted from G. An edge set $S\subseteq E\left(G\right)$ is called an edge cut-strategy if all the edges in S has been subverted from G and the survival subgraph, denoted by $G/S$, is disconnected, or is a single vertex, or is. The edge neighbor toughness of a graph G is defined to be $t_{EN}\left(G\right)=\underset{S\subseteq E\left(G\right)}{min}\left\{\frac{\left|S\right|}{c(G/S)}\right\}$, where S is any edge cut strategy of G, $c(G/S)$ is the number of the components of $G/S$. In this paper, the properties of this parameter are investigated, and the proof of the computation problem of edge neighbor toughness is $NP$-complete; finally, a polynomial algorithm for computing the edge neighbor toughness of trees is given.

1. Introduction

Gunther and Hartnell [1] introduced the idea of modeling a spy network by a graph whose vertices represent the spies and whose edges represent their connection. If a spy is arrested, the spies who are in direct contact with them are unreliable; therefore, some new graphical parameters such as vertex neighbor connectivity [1] and edge neighbor connectivity [2] were introduced to measure the invulnerability of networks under the “neighbor” case.

Observing that the behavior of spies in a spy network is similar to the spread of biological viruses in social networks, we introduced edge neighbor scattering number (ENS) [3] and vertex neighbor scattering number (VNS) [4]. It is shown that they are alternative invulnerability measures of the above networks. Since removing some vertices (or edges) from a graph, all of their adjacent vertices (or edges) are removed simultaneously, we call ENS and VNS neighbor invulnerability parameters.

Let $G=(V,E)$ be a graph and $e\in V\left(G\right)$. The open-edge neighborhood of e is defined $N\left(e\right)=\{f\in E(G\left)\right|f\ne e,e$ and f are adjacent}. The closed-edge neighborhood of e is $N\left[e\right]=N\left(e\right)\cup \left\{e\right\}$. An edge e is said to be subverted when $N\left[e\right]$ and the two endvertices of e are deleted from G. We call S an edge subversion strategy of G if $S\subseteq E\left(G\right)$ and each of the edges in S is 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 $G/S$ is disconnected, or is a single vertex, or is.

The edge neighbor scattering number of a connected graph G is defined as [3] $ENS\left(G\right)=\underset{S\subseteq E\left(G\right)}{max}\{c(G/S)-|S\left|\right\}$, where S is any edge cut strategy of G, and $c(G/S)$ is the number of the components of $G/S$. We call $S^{*}\subseteq E\left(G\right)$ a $ENS$-set of G if $ENS\left(G\right)=c(G/S^{*})-\left|S^{*}\right|$.

Inspired by the definitions of vertex neighbor connectivity and toughness, we define edge neighbor toughness(ENT) of a connected graph G as $t_{EN}\left(G\right)=\underset{S\subseteq E\left(G\right)}{min}\left\{\frac{\left|S\right|}{c(G/S)}\right\}$, where S is any edge cut strategy of G, and $c(G/X)$ is the number of the components of $G/S$. We call $S^{*}(\subseteq E\left(G\right))$ an $t_{EN}$-set of G if $t_{EN}\left(G\right)=\frac{|S^{*}|}{c(G/S^{*})}$.

In this paper, we prove that the computation problem of edge neighbor toughness of a graph is $NP$-complete. We also give a polynomial algorithm of the ENT of trees, which is a class of special and important graphs. Throughout this paper, we consider the simple, undirected graphs, and use Bondy and Murty [5] for terminologies and notations not defined here.

2. Preliminaries

Clearly, it is of prime importance to determine the edge neighbor toughness of a graph when this parameter is used to measure the neighbor invulnerability of a network. In this section, we give the edge neighbor toughness of several basic graphs.

Theorem 1.

Let $P_n$ be a path of order $n(\ge 3)$. Then

$$t_{EN}\left(P_n\right)=\left\{\begin{array}{cc}1,\hfill & ifn=3;\hfill \\ \frac{1}{2},\hfill & ifn\ge 4.\hfill \end{array}\right.$$

Proof. 

The case $n=3$ is trivial. When $n\ge 4$, for any edge cut strategy S, $c(P_n/S)\le \left|S\right|+1$. We thus have $t_{EN}\left(P_n\right)\ge \frac{\left|S\right|}{\left|S\right|+1}.$ Let $f\left(x\right)=\frac{x}{x+1}$ be a function of variate x, where $x\in Z_{+}$. When $x=1$, $f\left(x\right)$ reaches its minimum value $\frac{1}{2}$. So, we have $t_{EN}\left(P_n\right)\ge \frac{1}{2}$.

On the other hand, let $e=uv$ be an edge of $P_n$ such that $d\left(u\right)=d\left(v\right)=2$. Then $\left\{e\right\}$ is an edge cut strategy of $P_n$ and $c(P_n/\left\{e\right\})=2$. By the definition of ENT, we have $t_{EN}\left(P_n\right)\le \frac{\left|\right\{e\left\}\right|}{c(P_n/\left\{e\right\})}=\frac{1}{2}.$

Therefore, $t_{EN}\left(P_n\right)=\frac{1}{2}$.  □

Theorem 2.

Let $C_n$ be a cycle of order $n(\ge 3)$. Then

$$t_{EN}\left(C_n\right)=\left\{\begin{array}{cc}2,\hfill & ifn=4,5;\hfill \\ 1,\hfill & ifn=3 or n\ge 6.\hfill \end{array}\right.$$

Proof. 

$n=3,4,5$ is trivial. When $n\ge 6$, for any edge cut strategy S of $C_n$, $c(C_n/S)\le \left|S\right|$, we have $t_{EN}\left(C_n\right)\ge 1.$

On the other hand, there must exist two edges $e,f\in V\left(C_n\right)$ such that $\{e,f\}$ is an edge cut strategy of $C_n$ and $c(C_n/\{e,f\})=2$. By the definition of ENT, we have $t_{EN}\left(C_n\right)\le \frac{\left|\right\{e,f\left\}\right|}{c(C_n/\{e,f\})}=1.$

Therefore, $t_{EN}\left(C_n\right)=1$.  □

Theorem 3.

Let $K_n$ be a complete graph of order $n(\ge 3)$. Then $t_{EN}\left(K_n\right)=\lfloor \frac{n}{2}\rfloor$.

Proof. 

Observe that for any $e\in E\left(K_n\right)$, $K_n/\left\{e\right\}=K_{n-2}$. Cozzens [6] proved that the edge neighbor connectivity of $K_n$ is $\lfloor \frac{n}{2}\rfloor$; therefore, if S is an edge cut strategy of $K_n$, then $\left|S\right|\ge \lfloor \frac{n}{2}\rfloor$, and $c(K_n/S)\le 1$. By the definition of ENT, we have $t_{EN}\left(K_n\right)\ge \frac{\left|S\right|}{c(K_n/S)}\ge \lfloor \frac{n}{2}\rfloor$.

On the other hand, if n is even, let $M=\{u_1{v}_1,u_2{v}_2,\dots ,u_{\frac{n}{2}}{v}_{\frac{n}{2}}\}$ be a maximum matching of $K_n$. Replace $u_1{v}_1$ by $u_1{v}_2$, denote $M^{{}^{\prime }}=(M-\left\{u_1{v}_1\right\})\cup \left\{u_1{v}_2\right\}$. Then $K_n/M^{{}^{\prime }}=v_1$. By the definition of ENT, we have $t_{EN}\left(K_n\right)\le \frac{|M^{{}^{\prime }}|}{c(K_n/M^{{}^{\prime }})}=\lfloor \frac{n}{2}\rfloor$. If n is odd, then there exists a maximum matching M in $K_n$ such that $\left|M\right|=\frac{n-1}{2}=\lfloor \frac{n}{2}\rfloor$ and $K_n/M$ is an isolated vertex. So we have $t_{EN}\left(K_n\right)\le \lfloor \frac{n}{2}\rfloor$, too.

Therefore, $t_{EN}\left(K_n\right)=\lfloor \frac{n}{2}\rfloor$.  □

Theorem 4.

Let $K_{n_1,n_2,\dots ,n_p}$ be a complete p-partite graph with vertex partition $(X_1,X_2,\dots ,X_p)$. Assume $|X_i|=n_i,i=1,2,\dots ,p$, $p\ge 2$ and $n_p\ge n_{p-1}\ge \dots \ge n_1$. Then

$$t_{VN}\left(K_{n_1,n_2,\dots ,n_p}\right)=\frac{1}{n_p}(\sum _{i=1}^{p-1}{n}_i-{\alpha }^{{}^{\prime }}).$$

where ${\alpha }^{{}^{\prime }}$ is the matching number of $K_{n_1,n_2,\dots ,n_{p-1}}$.

Proof. 

For convenience, denote $K_{n_1,n_2,\dots ,n_p}$ as G. Obviously, for any edge cut strategy, S, of G, if $G/S\ne$, then there exists some i such that all the vertices of $G/S$ are included in $N_i$. Let $S^{*}$ be a $t_{EN}$-set of G. It is not difficult to know that $c(G/S^{*})=n_p$ and $S^{*}\subset E(G-X_p)$ such that $(G-X_p)/S^{*}=$.

Let $M^{*}$ be a maximum matching of $G-X_p$(that is, $K_{n_1,n_2,\dots ,n_{p-1}}$), and $U=\{u_1,u_2,\dots ,u_k\}$ be the unsaturated vertex set under $M^{*}$. Then $\{u_1{v}_1,u_2{v}_2,\dots ,u_k{v}_k\}\cup M^{*}$ is a $t_{EN}$-set of G, where $v_1,v_2,\dots ,v_k$ are arbitrary k vertices in $V\left(G\right)-X_p-U$. Since the unsaturated vertex number in $G-X_p$ is ${\displaystyle \sum _{i=1}^{p-1}}{n}_i-2\left|M^{*}\right|$, we have $t_{VN}\left(K_{n_1,n_2,\dots ,n_p}\right)=\frac{1}{n_p}({\displaystyle \sum _{i=1}^{p-1}}{n}_i-{\alpha }^{{}^{\prime }}).$ □

Remark 1.

If $K_{n_1,n_2,\dots ,n_{p-1}}$ has a perfect matching, then $t_{VN}\left(K_{n_1,n_2,\dots ,n_p}\right)=\frac{n}{2n_p}-\frac{1}{2}$, where $n={\displaystyle \sum _{i=1}^p}{n}_i$ and $n_p\ge n_{p-1}\ge \dots \ge n_1$.

Example 1.

Let $S_{1,n-1}$ be a star of order n. Then $t_{EN}\left(S_{1,n-1}\right)=\frac{1}{n-2}$.

A comet, denoted by $C_{n,k}$, is a graph by coinciding an end vertex of path $P_{n-k}$ with the center vertex of a star $S_{1,k}$, where $1\le k\le n-2$ and $n\ge 4$. The order of comet $C_{n,k}$ is n, and the center of $S_{1,k}$ is called the center of $C_{n,k}.$

Theorem 5.

Let $C_{n,k}$ be a comet, where $1\le k\le n-3$ and $n\ge 4$. Then $t_{EN}\left(C_{n,k}\right)=\frac{1}{k+1}$.

Proof. 

Let $V\left(P_{n-k}\right)=\{v_1,v_2,\dots ,v_{n-k}\}$, and $v_1$ be the center of $C_{n,k}$. Obviously, if S is an edge cut strategy of $C_{n,k}$, then $\left|S\right|\ge 1$. Moreover, for any edge cut strategy S, $c(C_{n,k}/S)\le k+\left|S\right|$ and the function $f\left(x\right)=\frac{x}{x+k}(x\in Z_{+})$ is decreased with x, we then have $t_{EN}\left(C_{n,k}\right)\ge \frac{1}{k+1}$.

On the other hand, since $\left\{v_1{v}_2\right\}$ is an edge cut strategy of $C_{n,k}$, and $c(C_{n,k}/\left\{v_1{v}_2\right\})=k+1$, by the definition of ENT, we have $t_{EN}\left(C_{n,k}\right)\le \frac{1}{k+1}$.

Therefore, $t_{EN}\left(C_{n,k}\right)=\frac{1}{k+1}$, and we complete the proof.   □

3. The Main Result

In this section, we consider the computational problems of ENT. We prove that the problem of computing the edge neighbor toughness is $NP$-complete and give a polynomial algorithm for computing the edge neighbor toughness of trees.

Problem 1.EDGE NEIGHBOR TOUGHNESS

Instance: An undirected graph G; a positive rational number t.

Question: Does there exist an edge cut strategy S of G such that $\frac{\left|S\right|}{c(G/S)}\le t$?

We solve this problem by considering the following

Problem 2.EDGE DOMINATION SET

Instance: A bipartite graph G; a positive integer d.

Question: Does there exist an edge dominating set D of G such that $\left|D\right|\le d$?

It was proved by Yannakakis and Gavril [7] that Problem 2 is NP-complete. Based on this conclusion, we prove that Problem 1 is NP-complete by reducing Problem 2 to a special case of Problem 1.

Theorem 6.

EDGE NEIGHBOR TOUGHNESS isNP-complete.

Proof. 

Clearly, EDGE NEIGHBOR TOUGHNESS is in the class NP, since a nondeterministic algorithm need only guess an edge cut strategy $S\subseteq E\left(G\right)$ and check in polynomial time that $\frac{\left|S\right|}{c(G/S)}\le t$.

Let $G=(V,E)$ be a bipartite graph of order n. Denote $V=\{v_1,v_2,\dots ,v_n\}$. Replace each vertex $v_i\in V$ by a copy of the complete graph $K_n$, denote this copy by $G_i$. Choose a vertex from $G_i$, $v_i^{*}$, add edges $v_i^{*}{v}_j^{*}$ if $v_i{v}_j\in E$, $i=1,2,\dots ,n$. Denote the resulting graph by $G^{*}$.

Denote the subgraph induced by $\{v_1^{*},v_2^{*},\dots ,v_n^{*}\}$ in $G^{*}$ as $G^{{}^{\prime }}$. Obviously, $G^{{}^{\prime }}\cong G$. Let D be a smallest edge dominating set of G and $S^{*}$ be an $ENS$-set of $G^{*}$. In [8], we proved the following claims.

Fact 1.If e is an edge in $G_i$, which is not incident with $v_i^{*}$, then $e\notin S^{*}$, $i=1,2,\dots ,n$.

Fact 2.Let $E_i^{*}=\{e:e\in E\left(G_i\right)$ and e be incident with $v_i^{*}$}. Then $|E_i^{*}\cap S^{*}|\le 1$, $i=1,2,\dots ,n$.

Fact 3.There exists a $t_{EN}$-set S of $G^{*}$ such that $E(G^{*}/S)\cap E\left(G^{{}^{\prime }}\right)=$ and $S\subseteq E\left(G^{{}^{\prime }}\right)$.

From the above claims, we conclude that $t_{EN}\left(G^{*}\right)=\frac{\left|D\right|}{n}$. By the construction of $G^{*}$ and the $NP$-completeness of Problem 2, this is sufficient for the conclusion.  □

Theorem 7.

Let T be a tree with order n (≥4). Then $t_{EN}\left(T\right)=\frac{1}{d-2}$, where $d=\underset{(u,v)\in E\left(T\right)}{max}\{d\left(u\right)+d\left(v\right)\}.$

Proof. 

Let $e^{*}=u^{*}{v}^{*}\in E\left(T\right)$ such that $d=d\left(u^{*}\right)+d\left(v^{*}\right)=\underset{(u,v)\in E\left(T\right)}{max}\{d\left(u\right)+d\left(v\right)\}.$ Since $n\ge 4$, $\left\{e^{*}\right\}$ is an edge cut strategy of T and $\frac{\left|\right\{e^{*}\left\}\right|}{c(T/\left\{e^{*}\right\})}=\frac{1}{d-2}$. Assume that $S\subset E\left(T\right)$ is any nonempty edge cut strategy of T. By the selection of $e^{*}$, we have

$$\frac{\left|S\right|}{c(T/S)}\ge \frac{\left|S\right|}{{\displaystyle \sum _{e\in S}}c(T/\left\{e\right\}}\ge \frac{\left|S\right|}{\left|S\right|\omega (T/\left\{e^{*}\right\})}=\frac{1}{d-2}.$$

Therefore, by the definition of ENT, $\left\{e^{*}\right\}$ is a $t_{EN}$-set of T, and $t_{EN}\left(T\right)=\frac{1}{d-2}$.

The proof is completed. □

Let T be a tree of order $n(\ge 4)$, $V\left(T\right)=\{v_1,v_2,\dots ,v_n\}$ and $M\left(T\right)={\left(m_{ij}\right)}_{n\times (n-1)}$ be its incident matrix. Then $d\left(v_i\right)={\displaystyle \sum _{j=1}^{n-1}}{m}_{ij},i=1,2,\dots ,n$.

Based on the above results, we give an algorithm for the problem of computing the ENT of trees.

Input A tree $T=(V;E)$.

Output$t_{EN}\left(T\right)$ and the corresponding $t_{EN}$-set.

Step 1. Compute the degree $d\left(v\right)$ for each vertex $v\in V\left(T\right)$;

Step 2. Compute $d_e=d\left(u\right)+d\left(v\right)$ for each edge $e=uv\in E\left(T\right)$;

Step 3. Search an edge $e^{*}=u^{*}{v}^{*}$ satisfying $d=d\left(u^{*}\right)+d\left(v^{*}\right)=\underset{e\in E\left(T\right)}{max}{d}_e$, output $t_{EN}\left(T\right)=\frac{1}{d-2}$ and a $t_{EN}$-set $\left\{u^{*}{v}^{*}\right\}$.

A graph-theoretic algorithm is good if the number of computational steps for its implementation on any graph is bounded above by a polynomial in the order and the size of the graph [5]. We show that the above algorithm is good by the following complexity analysis.

Computing the degree for each vertex requires $n-2$ additions, so the computations involved in step 1 require $n(n-2)$ additions. Computing $d_e=d\left(u\right)+d\left(v\right)$ for an edge $uv\in E\left(T\right)$ requires 1 additions only. Since T is a tree, $m=n-1$, step 2 requires $(n-1)$ additions. To find the edge $e^{*}=u^{*}{v}^{*}$, it is sufficient to sort the $n-1$ number $d_e$; therefore, step 3, in the worst case, requires $(n-1)log(n-1)$ comparisons (Quick Sort Algorithm).

It is well known that a comparison or an addition is a basic computational unit. So, the total number of computations of this algorithm is approximately $n(n-2)+(n-1)+(n-1)log(n-1)$, and thus is of order $n^2$.

Remark 2.

If the adjacency list is used instead of the incidence matrix, the complexity can be lowered to $O\left(n\right)$.

4. Invulnerability Analysis Based on ENT

Observe the definition of the neighbor invulnerability parameters such as ENS, VNS, and VNT, they all measure the state of a network after being most severely damaged. That is, the subversion strategy must be a cut strategy. Based on this common characteristic, we replace the subtraction in ENS by division to define the concept of ENT. By definition, the larger the ENT is, the more resilient the network is.

The following examples illustrate that the parameters ENT and ENS are independent, so they are well defined.

Example 2.

Consider the comet $C_{10,3}$ and $G_1$(see Figure 1a). They have the equal order 10, vertex neighbor connectivity 1, edge neighbor connectivity 1, and ENT $\frac{1}{4}$; however, $ENS\left(C_{10,3}\right)=3$, $ENS\left(G_1\right)=5$. Furthermore, $\left\{e\right\}$ is a $t_{EN}$-set of $G_1$ but is not an $ENS$-set; $\{e,f\}$ is an $ENS$-set of $G_1$ but is not a $t_{EN}$-set of $G_1$.

Figure 1. Two graphs with order 10.

Example 3.

Consider the Petersen graph $P(5,2)$(see Figure 1b) and the path $P_{10}$. They have the equal order 10 and ENS 1, but $t_{EN}\left(P(5,2)\right)=\frac{3}{4}$, $t_{EN}\left(P_{10}\right)=\frac{1}{2}$. The edge set $\{e,f,g\}$ is both of a $t_{EN}$-set and an $ENS$-set of $P(5,2)$.

5. Conclusions

The above two examples also show that the parameters ENT and ENS have their own characteristics and advantages when measuring the neighbor invulnerability of networks. Network invulnerability is an important problem in graph theory. Many parameters have been introduced to measure the relationship of invulnerability and structure of networks; however, there exist a lot of unresolved problems about the parameters computing [3,9,10]. Since trees have special structure and wide range of applications [11,12], the polynomial algorithm of ENT of trees has a certain theoretical and practical significance.

Observing the conclusion of Theorem 4, it is easy to know that the matching number of the complete p-partite graph $K_{n_1,n_2,\dots ,n_p}$ is determined by the numbers $n_1,n_2,\dots ,n_p$; therefore, to find a maximum matching of $K_{n_1,n_2,\dots ,n_p}$ is equivalent to divide $\{1,2,\dots ,p\}$ into two nonempty subset $N_1$ and $N_2$ such that $|{\displaystyle \sum _{i\in N_1}}{n}_i-{\displaystyle \sum _{j\in N_2}}{n}_j|$ is as small as possible. Later, we will consider this integer programming and related problems. Furthermore, the algorithm and the bound of ENT for graphs of general structure are also worth considering.