Computing the Edge-Neighbour-Scattering Number of Graphs
Zongtian Weia, Nannan Qib, and Xiaokui Yuec
a School of Science, Xi’an University of Architecture and Technology, Xi’an, Shaanxi 710055, P.R. China
b Science and Technology on EO-Control Laboratory, Luoyang Institute of Electro-Optic Equipment, Luoyang, Henan 471009, P.R. China
c School of Astronautics, Northwestern Polytechnical University, Xi’an, Shaanxi 710072, P.R. China
Reprint requests to Z. W.; E-mail:wzt6481@163.com
Z. Naturforsch.68a,599 – 604 (2013) / DOI: 10.5560/ZNA.2013-0059
Received January 10, 2013 / revised April 7, 2013 / published online October 2, 2013
A set of edgesXis subverted from a graphGby removing the closed neighbourhoodN[X]fromG.
We denote the survival subgraph byG/X. An edge-subversion strategyXis called an edge-cut strat- egy ofGifG/Xis disconnected, a single vertex, or empty. The edge-neighbour-scattering number of a graphGis defined asENS(G) =max{ω(G/X)− |X|:Xis an edge-cut strategy ofG}, where ω(G/X)is the number of components ofG/X. This parameter can be used to measure the vulnerabil- ity of networks when some edges are failed, especially spy networks and virus-infected networks. In this paper, we prove that the problem of computing the edge-neighbour-scattering number of a graph isNP-complete and give some upper and lower bounds for this parameter.
Key words:Vulnerability of Networks; Edge-Neighbour-Scattering Number; Computational Complexity;NP-Complete; Bounds.
1. Introduction
In this paper, we use [1] and [2] for terminology and notations not defined here and consider finite, simple, and undirected graphs only.
The concept of spy network was introduced by Gunther and Hartnell [3,4]. They modelled a spy net- work by a graph whose vertices represent the stations and whose edges represent the lines of communica- tion. The most important property of spy networks is that, if a station is destroyed, the adjacent sta- tions will be betrayed and so the betrayed stations be- come useless to the network as a whole. Therefore, instead of considering the vulnerability or invulnera- bility of a network in the classic sense, a number of other related parameters were introduced to deal with this circumstance, including vertex-neighbour connec- tivity [4], edge-neighbour connectivity [5], vertex- neighbour integrity [6], edge-neighbour integrity [7], vertex-neighbour-scattering number [8], and edge- neighbour-scattering number [9]. The common ground of these parameters is that, when removing some
vertices (or edges) from a graph, all of their adja- cent vertices (or edges) are removed. It is shown that these parameters have theoretical as well as applied significance in the design and analysis of networks such as spy networks and virus-infected networks, see [8,9].
LetG= (V,E)be a graph ande=uvbe an edge of G. The edgeeis said to besubvertedif the edgee, all of its incident edges, and the two ends ofe,uandv, are removed fromG[10]. A set of edgesX⊆Eis called an edge-subversion strategyofGif each of the edges inX has been subverted. The survival subgraph is denoted byG/X. An edge-subversion strategy X is called an edge-cut strategyofGifG/Xis disconnected, a single vertex, or empty.
LetG be a graph. The edge-neighbour connectiv- ityofG, denoted byΛ(G), is the minimum size of all edge-cut strategies ofG. Anedge-dominating set Dof Gis a set of edges such that every edge not inDis ad- jacent to an edge inD. Theedge-domination number ofGis defined to beγ0(G) =min{|D|:Dis an edge- dominating set ofG}.
© 2013 Verlag der Zeitschrift f¨ur Naturforschung, T¨ubingen·http://znaturforsch.com
The edge-neighbour-scattering number of a graph Gis defined as
ENS(G) =max{ω(G/X)− |X|:
Xis an edge-cut strategy ofG}, where ω(G/X) stands for the number of compo- nents ofG/X. We callX∗⊆E(G)anedge-neighbour- scattering set(ENS-set) ofGifENS(G) =ω(G/X∗)−
|X∗|.
The concept of edge-neighbour-scattering number was introduced in [9]. Some properties of this param- eter as well as some of its applications were discussed there when it is used to measure the vulnerability of networks. In this paper, we prove that the problem of computing the edge-neighbour-scattering number of a graph isNP-complete and give some upper and lower bounds of edge-neighbour-scattering number via some other well-known graphic parameters.
2. Computing Edge-Neighbour-Scattering Number is NP-Complete
It is of prime importance to determine the edge- neighbour-scattering number of a graph. In this sec- tion, we will investigate the complexity for computing the edge-neighbour-scattering number of a graph.
Problem 1. EDGE-NEIGHBOUR-SCATTERING NUMBER
Instance:A graphG; and an integerk.
Question:Does there exist an edge-cut strategyXofG such thatω(G/X)− |X| ≥k?
We solve this complexity problem by considering the following
Problem 2.EDGE-DOMINATION NUMBER Instance:A bipartite graphG; and a positive integerd.
Question:Does there exist an edge-dominating setD ofGsuch that|D| ≤d?
It was proved by Yannakakis and Gavril [11] that the problem EDGE-DOMINATION NUMBER isNP- complete. Based on this conclusion, we prove that the problem EDGE-NEIGHBOUR-SCATTERING NUM- BER is alsoNP-complete.
Theorem 1. EDGE-NEIGHBOUR-SCATTERING NUMBER isNP-complete.
Proof. LetG= (V,E)be a bipartite graph with ordern.
DenoteV ={v1,v2, . . . ,vn}. Replace each vertexvi∈ V by a copy of a complete graphKn, and denote this copy byGi. Select a vertex fromGi, and denote it by v∗i (i=1,2, . . . ,n). Add edgesv∗iv∗jifvivj∈E. Denote the resulting graph byG∗(An example ofGandG∗in casen=5 is shown in Figure1).
For convenience, denote the subgraph induced by {v∗1,v∗2, . . . ,v∗n} in G∗ as G0. Obviously, G0∼=G. As- sume that X∗ is an ENS-set of G∗, i.e., ENS(G∗) = ω(G∗/X∗)− |X∗|, andDis a smallest edge dominat- ing set ofG.
Clearly, EDGE-NEIGHBOUR-SCATTERING NUMBER is in the class NP. We now prove that ENS(G∗) = n− |D|. By the construction of G∗, and the NP-completeness of EDGE-DOMINATION NUMBER, this is sufficient for the conclusion.
Claim 1.Ifeis an edge inGiwhich is not incident with v∗i, thene6∈X∗,i=1,2, . . . ,n.
Proof. Otherwise, denoteX∗∗=X∗\ {e}. Notice that Gi/{e}=Kn−2andv∗i is still inGi/{e}. So we have ω(G∗/X∗) =ω(G∗/X∗∗). But |X∗∗|=|X∗| −1, thus ω(G∗/X∗∗)− |X∗∗|>ω(G∗/X∗)− |X∗|. This is con- tradictory to thatX∗is anENS-set ofG∗.
Claim 2.LetEi∗={e:e∈E(Gi)andeis incident with v∗i}. Then|Ei∗∩X∗| ≤1 fori=1,2, . . . ,n.
Proof. Suppose that, for somei,|Ei∗∩X∗| ≥2. With- out loss of generality, assume thate,f ∈Ei∗∩X∗ and e6= f. Denote X∗∗ =X∗\ {e}. Since Gi/{e,f}= Kn−3 and Gi/{f} = Kn−2, we have ω(G∗/X∗)≤ ω(G∗/X∗∗). But |X∗∗| = |X∗| −1, thus we have ω(G∗/X∗∗)− |X∗∗|>ω(G∗/X∗)− |X∗|. This is con- tradictory to thatX∗is anENS-set ofG∗.
v1 v2
v3 v4 v5
G
G ∗
v∗1 v∗2
v∗3 v∗4 v∗5
K5 K5
K5 K5 K5
Fig. 1. GraphsGandG∗.
Claim 3.There exists an ENS-setX of G∗such that E(G∗/X)∩E(G0) =/0 andX⊆E(G0).
Proof. Suppose thatX∗is anENS-set ofG∗such that E(G∗/X∗)∩E(G0)6= /0. Without loss of generality, we assumev∗iv∗j∈E(G∗/X∗)∩E(G0). Then any edge which is incident withv∗i orv∗jis not inX∗. By Claim 1, any edge of E(Gi)∪E(Gj)is not in X∗. Therefore, Gi,Gj andv∗iv∗j belong to one component of G∗/X∗. LetX∗∗=X∗∪ {v∗iv∗j}. Then we have
ω(G∗/X∗∗)≥ω(G∗/X∗) +1 and
ω(G∗/X∗∗)− |X∗∗| ≥ω(G∗/X∗) +1−(|X∗|+1)
=ω(G∗/X∗)− |X∗|.
On the other hand, sinceX∗∗is an edge-cut strategy of G∗, we have
ω(G∗/X∗∗)− |X∗∗| ≤ω(G∗/X∗)− |X∗|.
Thus
ω(G∗/X∗∗)− |X∗∗|=ω(G∗/X∗)− |X∗|=ENS(G∗).
This implies thatX∗∗is also anENS-set ofG∗. There- fore, if we add all the edges ofE(G∗/X)∩E(G0)toX, we then get anENS-setX ofG∗such thatE(G∗/X)∩ E(G0) =/0. In other words, there always exists anENS- setX ofG∗ such that all the edges ofG0are in X or adjacent to some edges ofX.
Let X∗ be an ENS-set of G∗. By Claims 1 and 2, we then need only to prove that Ei∗∩X∗=/0 for i=1,2, . . . ,n. Suppose that Ei∗∩X∗6=/0 for some i.
Assumeei∈Ei∗∩X∗. Then any edge inE(G0)which is incident with v∗i must be not inX∗. Otherwise, let X∗∗=X∗\ {ei}. Then we have
ω(G∗/X∗) =ω(G∗/X∗∗),|X∗∗|=|X∗| −1 and
ω(G∗/X∗∗)− |X∗∗|>ω(G∗/X∗)− |X∗|.
This is contradictory to thatX∗is anENS-set ofG∗. Claim 3 implies that, there exists an ENS-set X∗ of G∗such thatX∗ is also an edge dominating set of G0. Thus we have ω(G∗/X∗) =n, i.e., ENS(G∗) = ω(G∗/X∗)− |X∗|=n− |X∗|.
Note thatDis a smallest edge dominating set ofG andG0∼=G. So, the edge set corresponding toDinG0 is also a smallest edge dominating set ofG0. Therefore,
|X∗| ≥ |D|. We have
ENS(G∗) =n− |X∗| ≤n− |D|.
On the other hand, sinceDis a smallest edge dominat- ing set ofG, the edge set corresponding toDinG0is an edge-cut strategy ofG∗andω(G∗/X∗) =n. Thus we have
ENS(G∗)≥ω(G∗/D)− |D|=n− |D|.
Therefore, we haveENS(G∗) =n− |D|. The proof is complete.
3. Lower and Upper Bounds for Edge-Neighbour-Scattering Number
In this section, we give some lower and upper bounds for edge-neighbour-scattering number in terms of other well-known graphic parameters.
Theorem 2. Let G be a connected graph with order n>5, and M be a maximum but not perfect matching of G. Denote the set of the unsaturated vertices on M as V∗, and assume that δ∗=minv∈V∗{dG(v)}. Then ENS(G)≥2−δ∗.
Proof. Letw be a vertex inV∗such thatd(w) =δ∗. DenoteN(w) ={u1,u2, . . . ,uδ∗}and|M|=m. It is ob- vious thatm≥1. LetM∗={e:e∈M andeis inci- dent with at least one vertex inN(w)}. We then have
|M∗| ≤δ∗.
It is easy to know that|V∗| ≥1. For any edgeuv∈M and x,y∈V∗, it is impossible that both of xu∈E and vy∈E hold at the same time. Otherwise, there exists an M-augmenting path xuvy in G, i.e., M0= M\ {uv} ∪ {xu,yv}, which is a matching ofGgreater thanM, a contradiction.
On the other hand, no two vertices inV∗are adja- cent. If not, letxandybe two vertices inV∗such that xy∈E. ThenM∪ {xy}is a matching ofGgreater than M, contradicting to the choice ofM. In other words, every vertex ofN(w)is incident with one of the edges inMandM∗is an edge-cut strategy ofG.
We distinguish two cases forV∗as follows.
Case 1.|V∗| ≥2.
Obviously,ω(G/M∗)≥2. So we have ENS(G)≥ω(G/M∗)− |M∗| ≥2−δ∗.
Case 2.|V∗|=1.
Case 2.1.δ∗>m.
It is not difficult to know thatMis an edge-cut strat- egy ofG, so we have
ENS(G)≥ω(G/M)− |M|=1−m≥2−δ∗. Case 2.2.δ∗≤m.
Case 2.2.1.M∗6=M.
Sincen>5, we haveω(G/M∗)≥2 and ENS(G)≥ω(G/M∗)− |M∗| ≥2−δ∗. Case 2.2.2.M∗=M.
In this case, every vertex of N(w)is incident with exactly one edge of M, and vise versa. Therefore, d(w) =δ∗=m. It follows fromn>5 thatd(w)≥3.
Denote V0 =V(G)\N[w] ={v1,v2, . . . ,vm} and the subgraph induced byV0inGasG[V0].
Case 2.2.2.1.G[V0]is not a complete graph.
There exist two edges inM, sayuiviandujvjsuch thatui∈N(w),uj∈N(w)andvivj6∈E. DenoteM∗∗= (M\ {uivi,ujvj})∪ {wui,wuj}. Then we have|M∗∗|= mandω(G/M∗∗) =2. ThusENS(G)≥2−δ∗holds.
Case 2.2.2.2.G[V0]is a complete graph.
Denote M ={u1v1,u2v2, . . . ,umvm}. If there exist two verticesuiandvjsuch thati6=janduivj∈E. Let M0=M∪ {uivj} \ {uivi}. Then|M0|=δ∗andG/M0is a subgraph ofGwhich consists of two isolated vertices viandw. So we have
ENS(G)≥ω(G/M0)− |M0|=2−m=2−δ∗. If for anyi6= j,uivj6∈E. Suppose that there exist two vertices in N(w), sayuianduj, such thatuiuj6∈
E. LetX0=M\ {uivi,ujvj} ∪ {vivj,wuk}, wherek6=i andk6= j. Then |X0|=mandG/X0 is a subgraph of G which consists of two isolated vertices ui and uj. Therefore, we have
ENS(G)≥ω(G/X0)− |X0|=2−m=2−δ∗.
Suppose that any two vertices in N(w) are adja- cent. Denote bm2c=k. When m is even, let X00 = {u1u2,u3u4, . . . ,um−1um}. We have|X00|=k=m2 and ω(G/X00) =2. Therefore,
ENS(G)≥ω(G/X00)− |X00|=2−k>2−δ∗. When m is odd, let X00 = {u1u2,u3u4, . . . , u2k−1u2k,umvm}. We haveω(G/X00) =2 and|X00|<m.
Therefore,
ENS(G)≥ω(G/X00)− |X00|>2−m>2−δ∗. The proof is complete.
Remark 1. The lower bound in Theorem2is best pos- sible. For example, whenn≥7 andnis odd, we have δ∗=2 andENS(Cn) =0.
Theorem 3. Let G be a graph with order n≥3 and γ0(G) be the edge domination number of G. Then ENS(G)≥max{1−γ0(G),n−3γ0(G)}.
Proof. The cases n=3, 4 are trivial. Suppose that n≥5 and Dis a smallest edge dominating set of G.
Obviously,D is an edge-cut strategy of G. Let G[D]
be the subgraph induced byDinG. Then|V(G[D])| ≤ 2γ0(G). By the definition of edge-dominating set, we know thatG/Dis empty or consists of isolated ver- tices.
Case 1.G/D6=/0.
It is easy to see that there are at least n−2γ0(G) isolated vertices inG/D. So we have
ENS(G)≥ω(G/D)− |D| ≥n−2γ0(G)−γ0(G)
=n−3γ0(G).
On the other hand, ifG/D6=/0, thenω(G/D)≥1. Thus ENS(G)≥ω(G/D)− |D| ≥1−γ0(G).
So we have
ENS(G)≥max{1−γ0(G),n−3γ0(G)}. Case 2.G/D=/0.
Each vertex ofV(G)is incident with an edge ofD.
Assume that uv∈D. Then it is impossible that both uandvare incident with some edges ofDexceptuv.
Otherwise, D\ {uv} is an edge-dominating set of G
smaller thanD, a contradiction. Sincen≥3, there must exist an edgee∈Dsuch thatN(e)∩(E(G)\D)6=/0. Let D∗= (D\ {e})∪ {f}, where f is an arbitrary edge of N(e)∩(E(G)\D). Then|D∗|=γ0(G)andG/D∗is an isolated vertex. So we have
ENS(G)≥ω(G/D∗)− |D∗|=1−γ0(G).
On the other hand, sinceG[D]is a spanning subgraph ofG, we have|V(G[D])|=n≤2γ0(G). Therefore
1−γ0(G)−(n−3γ0(G)) =2γ0(G)−n+1>0. Therefore,
ENS(G)≥1−γ0(G) =max{1−γ0(G),n−3γ0(G)}. The proof is complete.
Remark 2. The lower boundn−3γ0(G)in Theorem3 is best possible. LetCnbe the cycle with ordern(≥6) and n ≡0 (mod 3). Then we have γ0(Cn) = n3 and ENS(Cn) =0=n−3γ0(Cn). On the other hand, al- thoughC3 attains the bound 1−γ0(G), we have not found general examples to illustrate that this bound is best possible.
Theorem 4. Let G be a connected graph with order n and α0(G) be the matching number of G. Then ENS(G)≥n−3α0(G).
Proof. Assume thatM is a maximum matching ofG.
ThenMis an edge-cut strategy ofG. IfGhas a perfect matching, then we have
|M|=α0(G) =n
2, G/M=/0, and
ENS(G)≥ω(G/M)− |M|=−α0(G) =−n 2. The conclusion holds.
IfGhas no perfect matchings, thenG/Mconsists of only isolated vertices, andω(G/M) =n−2α0(G). We have
ENS(G)≥ω(G/M)− |M|=n−2α0(G)−α0(G)
=n−3α0(G). The proof is complete.
Remark 3. The lower bound in Theorem4is best pos- sible. For example, the complete graphs with odd order achieve this bound.
In the following, we give two upper bounds for the edge-neighbour-scattering number.
Theorem 5. Let G be a connected graph with order n andΛ(G)be the edge-neighbour-connectivity of G.
Then ENS(G)≤n−1−2Λ(G).
Proof. Let X be an edge-cut strategy ofG. Since X subverted fromGmeans deleting at least|X|+1 ver- tices ofG, we have
ENS(G)≤n−(|X|+1)− |X| ≤n−1−2Λ(G).
Corollary 1. Let G be a connected graph with order n≥3. Then ENS(G)≤n−3.
Remark 4. The upper bound in Theorem 5 is best possible. The stars and double stars can achieve this bound.
4. Conclusions and Future Research
In this paper, we prove that the problem of comput- ing the edge-neighbour-scattering number of a graph is NP-complete and give some upper and lower bounds for this parameter. Here we list some other related in- teresting research problems.
Harary [12] determined the maximum and mini- mum connectivity of graphs with given order and size and also constructed corresponding extremal graphs, which are now widely known as the Harary graphs.
Since then, finding the maximum or minimum value of graphic parameters with given order and size has become an attractive topic in graph theory. The oppo- site problem is finding the maximum or minimum size (order) when some parameters are given. It is natural to consider these two types of problems for the edge- neighbour-scattering number.
A Nordhaus–Gaddum type result is a (tight) lower or upper bound for the sum or product of the val- ues of a parameter for a graph and its complement.
Since Nordhaus and Gaddum [13] got the first result of this type on the chromatic number of graphs, many other similar results have been obtained (see [14] for a survey). It is interesting to investigate the Nordhaus–
Gaddum type result for the edge-neighbour-scattering number.
As we have shown, the problem of computing the edge-neighbour-scattering number of a graph is NP- complete, so it is interesting to consider whether we can find polynomial algorithms for computing this pa- rameter of some special classes of graphs.
Acknowledgements
This work was supported by NSRP (Grant No.
2012JC2-03) and ESF (Grant No. 12JK0888). The au- thors are grateful to the anonymous referees for valu- able comments and suggestions on the earlier versions of this article.
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