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Domination in Cayley graphs: A survey
T. Tamizh Chelvam
∗, M. Sivagami
Department of Mathematics, Manonmaniam Sundaranar University, Tirunelveli 627012, India
Received 1 May 2017; received in revised form 22 November 2017; accepted 22 November 2017 Available online xxxx
Abstract
LetΩbe a symmetric generating set of a finite groupΓ. Assume that (Γ,Ω) be such thatΓ = ⟨Ω⟩andΩsatisfies the two conditionsC1: the identity elemente̸∈ΩandC2: ifa∈Ω, thena−1∈Ω.Given (Γ,Ω) satisfyingC1andC2,define aCayley graph G=Cay(Γ,Ω) withV(G)=ΓandE(G)= {(x,y)a|x,y∈Γ,a∈Ωandy=xa}. WhenΓ=Zn= ⟨Ω⟩, it is called as circulant graphand denoted byCir(n,Ω). In this paper, we give a survey about the results on dominating sets in Cayley graphs and circulant graphs.
c
⃝2017 Kalasalingam University. Publishing Services by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
Keywords:Cayley graph; Unitary Cayley graph; Dominating sets; Efficient dominating sets; E-Chain
1. Introduction
The concept of Cayley graph plays a vital role in dealing with certain optimization problems, especially routing problems in interconnection networks of a parallel computer. Parallel processing and super computing continue to exert great influence in the development of modern science and engineering. The network comprising of processors plays a vital role in facilitating the communication between processors in a computer system. Some of the popular interconnection schemes are rings, torus and hypercubes. Their popularity stems from the commercial availability of machines with these architectures. These three families of graphs – rings, torus and hypercube – share a common property of being a Cayley graph. Many important problems in networks have been modeled by Cayley graphs.
Since Cayley graphs have the property of vertex transitivity, it is possible to implement routing and communication schemes at each node of the network. One of the principal issues concerning routing problems is identification of perfect dominating sets in Cayley graphs. This will help in identifying optimal substructures in order to facilitate the communication between processors.
Cayley graph is a discrete structure created out of groups, more specifically from a finite groupΓ and its generating setΩ.A non-empty subsetΩ ⊂Γ is called agenerating set forΓ, denoted byΓ = ⟨Ω⟩, if every element ofΓ can
Peer review under responsibility of Kalasalingam University.
∗ Corresponding author.
E-mail addresses:[email protected](T. Tamizh Chelvam),[email protected](M. Sivagami).
https://doi.org/10.1016/j.akcej.2017.11.005
0972-8600/ c⃝2017 Kalasalingam University. Publishing Services by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
be expressed as a product of elements inΩ. For a generating setΩofΓ, we assume that C1: The identity elemente̸∈ΩandC2: Ifa ∈Ω, thena−1∈Ω.
The concept of domination in graphs appears as a natural model for facility location problems, and has many applications in design and analysis of communication networks, network routing and coding theory, among others [1].
Another application of dominating sets is broadcasting in wormhole-routed networks. Dominating sets have many uses in design theory and efficient use in computer networks. They can be used to decide the placement of limited resources so that every node has access to the resources locally or through a neighboring node. A minimum dominating set provides this access to resources at minimum cost. The most efficient solution is one that avoids duplication of access to the resources. This more restricted version of minimum dominating set is called an independent perfect or efficient dominating set. For basic definitions and results concerning domination in graphs, one can refer to [1]. In this paper, we make a survey of results connecting the domination in Cayley graphs, domination, connected domination, total domination and efficient domination on circulant graphs. Finally, we present results connecting subgroups of a group as efficient dominating sets in the corresponding Cayley graphs.
2. Domination in Cayley graphs
Even though Cayley graphs are extensively dealt in various literature, only few authors have worked on domination in Cayley graphs. To understand the concept of domination for Cayley graphs, one can refer to [2–6]. Lee [6]
has studied the efficient dominating sets in Cayley graphs using covering projections. He obtained a necessary and sufficient condition for the existence of an efficient dominating set in a Cayley graph. As an application, he classified the hypercubes which admit efficient dominating sets. In this section, we present results related to domination in Cayley graphs.
Lemma 2.1([6, Lemma 1]).
(a) Let S1and S2be two independent perfect dominating sets of a graph G. Then|S1| = |S2|.
(b) Let S1, . . . ,Sn be n independent perfect dominating sets of a graph G which are pairwise mutually disjoint.
Then the subgraph H induced by S1∪ · · · ∪Snis an m-fold covering graph of the complete graph Kn, where m= |Si|for each i =1,2, . . . ,n.
Lemma 2.2([6, Lemma 2]). Let p:G˜→G be a covering and let S be a perfect dominating set of G. Then p−1(S) is a perfect dominating set of G. Moreover, if S is independent, then p˜ −1(S)is independent.
Theorem 2.3([6, Theorem 1]). Let G be a graph and let n be a natural number. Then G is a covering of the complete graph Kn if and only if G has a vertex partition{S1, . . . ,Sn}such that Si is an independent perfect dominating set for each i=1,2, . . . ,n.
Lemma 2.4([6, Lemma 3]). Let X = {x1, . . . ,xn}be a symmetric generating set for a groupAand let S be an independent perfect dominating set of the Cayley graph Cay(A,X). Then
(a) for each i=1,2, . . . ,n, xiS is an independent perfect dominating set of Cay(A,X), and (b) {S,Sx1, . . . ,Sxn}forms a vertex partition of Cay(A,X).
For a subsetSof a groupA, defineS0=S∪ {0}, where 0 is the identity element inA.
Theorem 2.5([6, Theorem 2]). Let X = {x1, . . . ,xn}be a symmetric generating set for a groupAand let S be a normal subset ofA. Then the following are equivalent.
(a) S is an independent perfect dominating set of Cay(A,X).
(b) There exists a covering p:Cay(A,X)→ K|X|+1such that p−1(v)=S for somev∈V(K|X|+1).
(c) |S| = |A|
|X|+1 and S∩[S+((X◦+X◦)\ {0})]= ∅.
Theorem 2.6([6, Corollary 2]). Let X = {x1, . . . ,xn}be a symmetric generating set for a groupAand let S be a normal subgroup of A. Then the following are equivalent.
(a) S is an independent perfect dominating set of Cay(A,X).
(b) Cay(A,X)is an S-covering graph of the graph K|X|+1. (c) |A/S| = |X| +1and S∩(X◦+X◦)= {0}.
Theorem 2.7([6, Theorem 3]). Let n be a natural number. Then the following are equivalent.
(a) The hypercube Qnhas an independent perfect dominating set.
(b) n=2m−1for a natural number m.
(c) The hypercube Qnis a regular covering of the complete graph Kn+1.
Italo J. Dejter and Oriol Serra [2] gave a constructing tool to produce E-chains of Cayley graphs. This tool is used to construct infinite families of E-chains of Cayley graphs on symmetric groups.
Lemma 2.8([2, Lemma 1]). Let A be a generating set of a finite group G such that s2=1for each s∈ A. Let u∈ A be such that Au =A\ {u}generates a proper subgroup H of G of index|A| +1in G. If U =H∩u H u is an E -set in Cay(H,Au), then the open neighborhood N(H)of H is an E -set in Cay(G,A). Moreover, there are inclusive maps ς(j)such that{ς(j)(U),j=1, . . . ,|A|}is a partition of the E -set N(H).
Further, Italo J. Dejter [5] obtained a necessary and sufficient condition for the existence of an efficient open dominating set (1-perfect code) in Cartesian product of two cycles.
Theorem 2.9([5, Theorem 7]). There exists a toroidal graph Cm×Cnhaving a 1-perfect code partition if and only if m and n are multiples of 5.
Hamed Hatami and Pooya Hatami [3] characterized the structure of perfect dominating sets in the simplest caseCay(Zn2n+1;U) where 2n +1 is a prime,U = {±e1, . . . ,±en} is the set of generators ofZn2n+1 andei = (0, . . . ,1, . . . ,0) is the unit vector with 1 at theith coordinate.
Theorem 2.10([3, Theorem 1]). Let2n+1be a prime and S⊆Cay(Zn2n+1,U)be a perfect dominating set. Then for every(x1, . . . ,xn)∈Zn2n+1and every i ∈ {1, . . . ,n},|S∩{
(y1, . . . ,yn):yj =xj∀j ̸=i}
| =1.
Circulant graphs are Cayley graphs on the simplest of groups and thus may provide direction into Cayley graphs on other groups, particularly finite groups. Any finite vertex-transitive graph of prime order is a circulant graph and hence the study of vertex transitive graphs can gain from the study of circulant graphs. Nenad Obradoviˇc, Joseph Peters, and Ruˇzi´c [7] studied the efficient domination in circulant graphs with two chord lengths. They have obtained necessary and sufficient conditions for the existence of an efficient dominating set in circulant graphs of degree 3 and 4. The following are some of the important results concerning domination in circulant graphs.
Theorem 2.11([7, Theorem 1]). Let G=Cir(n,S = {s1,s2})be a connected 4-regular circulant graph. G admits efficient domination if and only if n=5.i,i ∈Nand|s1±s2| ̸=0(mod5)and s1,s2̸=0(mod5).
Theorem 2.12([7, Theorem 2]). Let G =Cir(n,S = {s1,s2(= n
2)})be a connected 3-regular circulant graph. G admits efficient domination if and only if n=8j+4for some j ∈N.
Jia Huang and Jun-Ming Xu [4] have studied the relationship between the bondage number and the efficient dominating sets of vertex-transitive graphs. Further, they have proved that some Harary graphs admit efficient dominating sets. Letk <n. The Harary graphHk,nis a graph of ordernand connectivitykwith minimum number of edges. Lets = k
2 whenkis even ands = k−1
2 whenkis odd. Ifkis even and A = {1,2, . . . ,s,n−s,n−(s− 1), . . . ,n−1}, then the corresponding circulant graphCir(n,A) is denoted by Hk0,n [4]. Similarly ifkis odd,n is even andA= {1,2, . . . ,s,n2,n−s,n−(s−1), . . . ,n−1}, thenCir(n,A) is denoted byHk1,n[4].
Lemma 2.13([4, Lemma 4.5]). γ(Hk0,n)= ⌈ n
k+1⌉.
Lemma 2.14([4, Lemma 4.7]). Let G= Hk1,n. Then G has an efficient dominating set if and only if n=(k+1)p for an odd p, and all efficient dominating sets in G have the form Di= {v∈V(G):v≡i(mod k+1)}.
Lemma 2.15([4, Lemma 4.3]). Let G=Cir(n; {1,s})with s̸= n
2. Then⌈n
5⌉ ≤γ(G)≤ ⌈n
3⌉, and G has an efficient dominating set if and only if 5|n and s ≡ ±2(mod 5). In addition, all efficient dominating sets in G have the form Di= {v∈V(G):v ≡i(mod5)}.
Tamizh Chelvam and Mutharasu [8] characterized some classes of circulant Harary graphs which admit efficient dominating sets and the same are presented below.
Lemma 2.16([8, Lemma 9]). If 3s+1divides n, thenγt(Hk0,n)= 2n
3s+1.
Theorem 2.17([8, Theorem 10]). The graph Hk0,n has an efficient open dominating set if and only if k =2and4 divides n.
Lemma 2.18([8, Lemma 11]). If 4s+2divides n, thenγt(Hk1,n)= 2n
4s+2.
Theorem 2.19([8, Theorem 12]). The graph Hk1,nhas an efficient open dominating set if and only if4s+2divides n.
A different approach was made by Young Soo Kwon and Jaeun Lee [9] in dealing the perfect dominating sets in Cayley graphs. They show that if a Cayley graphCay(A,X) has a perfect dominating set S which is a normal subgroup ofAand whose induced subgraph isF, then there exists anF-bundle projectionp:Cay(A,X)→ Kmfor some positive integerm. As an application, they studied perfect dominating sets in the hypercubeQn. It is known that Qn is isomorphic to the Cayley graphCay(Zn2,X), whereX = {e1,e2, . . . ,en}.
Theorem 2.20([9, Theorem 7]). Let X = {x1, . . . ,xn}be a symmetric generating set for a group Aand let S be a normal subgroup of A. Let S∩ X = {xℓ,xℓ+1, . . . ,xn}and letBbe the subgroup generated by S∩X . Let F = |S/B|Cay(B,S ∩ X) be the disjoint union of |S/B| copies of the Cayley graph Cay(B,S ∩ X). Now the following are equivalent.
(a) S is an F -perfect dominating set of Cay(A,X).
(b) There exists an F -bundle projection p : Cay(A,X)→ Kℓ such that p−1(vℓ) = S, where Kℓis the complete graph onℓverticesv1, v2, . . . , vℓ.
(c) |A/S| =ℓand S∩(Xℓ−1)2= {1}, where1is the identity and Xℓ−1= {x1,x2, . . . ,xℓ−1}, and A2= {aa′|a,a′∈ A}for any subset A of A.
LetAbe an abelian group and letSbe a subgroup ofA. For any symmetric generating setX = {x1, . . . ,xn}ofA, letS∩X = {xℓ,xℓ+1, . . . ,xn}. LetBbe the subgroup generated byS∩X and letX =X/B= {x1, . . . ,xℓ−1}. Now Xis a symmetric generating set for the quotient groupA/B. The following corollary comes fromTheorem 2.20.
Corollary 2.21([9, Corollary 8]). Let X = {x1, . . . ,xn}be a symmetric generating set for an abelian groupAand let S be a subgroup ofA. Then the following are equivalent.
(a) S/Bis an efficient dominating set of Cay(A/B,X).
(b) There exists a regular covering projection p:Cay(A/B,X)→ Kℓsuch that p−1(vℓ)=S/B. (c) |(A/B)/(S/B)| = |A/S| =ℓand S/B∩X2 = {1}, where1is the identity of A/B.
The following theorem gives several necessary and sufficient conditions for a hypercubeQnto have a perfect total dominating set.
Theorem 2.22([9, Theorem 11]). For a positive integer n, the following are equivalent.
(a) The hypercube Qnhas a perfect total dominating set.
(b) n=2mfor a positive integer m.
(c) The hypercube Qnis a2n−log2n−1K2-bundle over the complete graph Kn. (d) The hypercube Qnis a covering of the complete bipartite graph Kn,n.
Corollary 2.23([9, Corollary 12]). Let m and n be two positive integers such that n−m+1is a power of 2. Then the hypercube Qn is a2n−m−log2(n−m+1) Qm-bundle over the complete graph Kn−m+1 and hence has a2n−m−log2(n−m+1) Qm- perfect dominating set.
3. Efficient open domination in Cayley graphs
In this section, we present results in connection with bipartite Cayley graphs which admit efficient open dominating sets.
Lemma 3.1([8, Lemma 4]). Let X = {x1,x2, . . . ,xn}be a generating set of a groupΓ and S be an efficient open dominating set of G=Cay(Γ,X). Then we have the following:
(a) For each i with1≤i≤n, xiS is an efficient open dominating set in G.
(b) {Sx1,Sx2, . . . ,Sxn}is a vertex partition of G.
Lemma 3.2([8, Lemma 5]). Let S1,S2, . . . ,Snbe pairwise disjoint efficient open dominating sets of a graph G and G be the subgraph of G, induced by S˜ 1∪S2∪ · · · ∪Sn. Let m = |S1|
2 . If G is bipartite, then there exists a m-fold˜ covering projection fromG onto K˜ n,n.
FromLemmas 3.1and3.2, we have the following corollary.
Corollary 3.3([8, Corollary 1]). Let G=Cay(Γ,X)be a bipartite Cayley graph with X= {x1,x2, . . . ,xn}and let S be an efficient open dominating set of G. If xiS=Sxifor each i =1, . . . ,n, then there exists a covering projection f :G→ Kn,n such that Sxi = f−1({yi,zi})for1≤i ≤n, where V(Kn,n)=(Y,Z)with Y = {y1,y2, . . . ,yn}and Z = {z1,z2, . . . ,zn}.
Lemma 3.4([8, Lemma 6]). Let f :G˜→G be a covering projection and S be an efficient open dominating set in G. Then f−1(S)is an efficient open dominating set inG.˜
Theorem 3.5([8, Theorem 7]). A graph G is a covering graph of Kn,nif and only if G is bipartite and has a vertex partition{S1,S2, . . . ,Sn}such that Si is an efficient open dominating set in G for1≤i ≤n.
Theorem 3.6([8, Theorem 8]). Let G =Cay(Γ,X)be a bipartite Cayley graph with X = {x1,x2, . . . ,xn}and let S be a normal subset of Γ (i.e., g S=Sg for all g ∈Γ). Then the following are equivalent.
(a) S is an efficient open dominating set of G.
(b) There exists a covering projection f : G → Kn,n such that f−1({yi,zi}) = S for some1 ≤ i ≤ n, where V(Kn,n)=(Y,Z)with Y = {y1,y2, . . . ,yn}and Z = {z1,z2, . . . ,zn}.
(c) |S| = |Γ|
n and S∩[S((X X)\ {e})]=φ, where e is the identity of Γ.
Assume thatk <n. Consider a graph of ordern and connectivitykwith minimum number of edges. Ifkis odd andnis even, letm= n
2 −1. IfX = {m−(p−1), . . . ,m−1,m,n2,n−m,n−(m−1), . . . ,n−(m−(p−1))}, thenGk,ndenotes the circulant graphCir(n,X).
Lemma 3.7([8, Lemma 13]). γt(Gk,n)=⌈n k
⌉.
Theorem 3.8([8, Theorem 14]). The graph Gk,nhas an efficient open dominating set if and only if k divides n.
A countable family of graphsG = {Γ1⊆Γ2⊆. . .⊆Γi ⊆Γi+1⊆. . .}is an E-chain if every Γi is an induced subgraph ofΓi+1 and eachΓi has an efficient dominating setCi [2]. Dejter et al. [2] derived infinite families of E-chains in Cayley graphs constructed from symmetric groups. Using this, one can obtain an efficient dominating set for a Cayley graph of order (n+1)!from an efficient dominating set of a Cayley graph of ordern!. The difference between n! and (n +1)! is large and one needs to find some tool to obtain efficient dominating sets in Cayley graphs of ordert, wheren! < t < (n +1)!. With this in mind, Tamizh Chelvam and Mutharasu [8] constructed E-chains in circulant graphs using covering projections. Letn and p be positive integers with 1 ≤ p ≤ n−1
2 . Let X ={
x1,x2, . . . ,xp,n−xp,n−xp−1, . . . ,n−x1
}⊆Znwithgcd(x1,x2, . . . ,xp)=1. Note thatXis a generating set of the groupZn.
Lemma 3.9 ([8, Lemma 15]). Let n,q ≥ 2 and p ≥ 1 be integers with 1 ≤ p ≤ n−1
2 . Let X = {x1,x2, . . . ,xp,n−xp,n−xp−1, . . . ,n−x1
}and Y = {
x1,x2, . . . ,xp,nq−xp,nq−xp−1, . . . ,nq−x1
}. Then G˜=Cir(nq,Y)is a covering graph of G=Cir(n,X).
ByLemmas 2.2,3.4and3.9, one can have the following theorem, which gives E-chains and EO-chains in circulant graphs.
Theorem 3.10([8, Theorem 16]). Let n,qi ≥2be integers for i =1,2, . . .and S be an efficient open dominating set (or efficient dominating set) of Cir(n,X),where X ={
x1,x2, . . . ,xp,n−xp,n−xp−1, . . . ,n−x1
}. Suppose Xi ={
x1,x2, . . . ,xp,nq1q2. . .qi−xp,nq1q2. . .qi−xp−1, . . . ,nq1q2. . .qi−x1
}for i = 1,2, . . .. Then an EO- chain (or E-chain) for the graphs Cir(n,X), Cir(nq1,X1), Cir(nq1q2,X2), . . . ,is given by S ⊂ S1 ⊂ S2 ⊂S3 ⊂ . . . ,where S1,S2, . . .are constructed as follows:
(a) S1= fq−1
1 (S), where fq1 :Cir(nq1,X1)→Cir(n,X)is a covering projection and (b) Si = fq−1
i (Si−1), where fqi : Cir(nq1q2. . .qi,Xi) → Cir(nq1q2. . .qi−1,Xi−1)is a covering projection for each integer i ≥2.
4. Domination in circulant graphs
Efficient dominating sets correspond to perfect 1-correcting codes in a network. It makes the study of dominating sets and efficient dominating sets in circulant graphs as an important one. To study about domination in circulant graphs, one may refer [4,7]. In [4], Jia Huang and Jun-Ming Xu obtained the domination number for some classes of circulant graphs. Further, they identified some classes of circulant graphs which admits an efficient dominating set. In [7], Nenad Obradoviˇc, Joseph Peters, and Ruˇzi´c studied the efficient dominating sets in circulant graphs with two chord lengths. In this section, we present results connecting domination, total domination, connected domination, perfect domination, independent domination and efficient domination numbers for some circulant graphs.
The domination number depends on the elements in the generating set, and so it is very difficult to find an algorithm to obtain aγ-set for an arbitrary circulant graph. Finding the domination number for an arbitrary circulant graph is still an unsolved problem. Tamizh Chelvam, Rani and Mutharasu [8,10–15] have obtained the domination number for some classes of circulant graphs. Further, they proved that the respective class of circulant graphs are 2-excellent.
Unless otherwise specified, Astands for the generating set{1,2, . . . ,k,n−k,n−(k−1), . . . ,n −1}ofZn with 1≤k≤ n−1
2 andBstands for the generating set{1,3, . . . ,2k−1,n−(2k−1), . . . ,n−3,n−1}with 1≤k≤ n−1
2
and (2k−1)̸=n−(2k−1) (to ensure that the elements ofBare distinct). Tamizh Chelvam and Rani [10] obtained the value of the domination number ofCir(n,A), whereA= {1,2, . . . ,k,n−k, . . . ,n−1}. They identified aγ-set forCir(n,A) and having identified aγ-set, they found certain otherγ-sets inCir(n,A) by using the inverse property of the underlying group. AmongCir(n,A), they characterized all 2-excellent circulant graphs. In [11], they obtained γ γ-minimum, domatic number, independent domatic number, perfect domatic number and anE-set in these classes of circulant graphs. Further, they identified some circulant graphs which are having the properties ofγi-excellent, just excellent and domatically full.
Theorem 4.1([10, Theorem 2.1]). Let n(≥3)and k be positive integers such that1≤k≤ n−1
2 . Let G =Cir(n,A), where A= {1,2, . . . ,k,n−k, . . . ,n−2,n−1}. Thenγ(G)= ⌈ n
|A|+1⌉. Remark 4.2([10, Remark 2.5]). Ifn(≥3) is an even integer and|A| ≥ n
2, thenγ(Cir(n,A))= ⌈ n
|A|+1⌉ ≤ ⌈nn
2+1⌉ ≤
⌈2n
n+2⌉ ≤2. Ifnis odd and|A| ≥ ⌊n
2⌋, thenγ(Cir(n,A))= ⌈ n
|A|+1⌉ ≤2.
Theorem 4.3([10, Theorem 2.9]). Let n(≥3)and k be positive integers such that1≤k≤ n−1
2 . Let G =Cir(n,A), where A= {1,2, . . . ,k,n−k, . . . ,n−2,n−1}. Then the graph G =Cir(n,A)is excellent.
Theorem 4.4([10, Theorem 2.10]). Let n(≥3)and k be positive integers such that1≤k≤ n−1
2 . Let G=Cir(n,A), where A= {1,2, . . . ,k,n−k, . . . ,n−2,n−1}. Suppose n=(2k+1)t+1for some positive integer t , then G is 2-excellent.
Lemma 4.5([10, Lemma 2.7]). Let n(≥3), k be integers such that1≤k≤ n−1
2 andℓ= ⌈ n
|A|+1⌉. Let G=Cir(n,A), where A = {1,2, . . . ,k,n −k, . . . ,n −1}. If n = (2k+1)(ℓ−1)+h, for some h with 1 ≤ h ≤ 2k, then D= {0,h,h+(2k+1),h+2(2k+1), . . . ,h+(ℓ−2)(2k+1)}is aγ-set for G.
Theorem 4.6 ([13, Theorem 2.5]). Let n(≥ 3), k be integers such that 1 ≤ k ≤ n−1
2 and ℓ = ⌈ n
2k+1⌉. Let A = {1,2, . . . ,k,n −k, . . . ,n −1}and n = (ℓ−1)(2k+1)+ j with1 ≤ j ≤ 2k. Then G = Cir(n,A) is 2-excellent if and only if j =1.
Lemma 4.7([16, Lemma 3.2.12]). Let n(≥ 3), k be integers such that1 ≤ k ≤ n−1
2 and G = Cir(n,A), where A= {1,2, . . . ,k,n−k, . . . ,n−1}. Then D+h is aγ-set for any positive integer h with1≤h ≤n−1, whenever D is aγ-set of G.
Lemma 4.8([16, Lemma 3.2.13]). Let n(≥ 3), k be integers such that1 ≤ k ≤ n−1
2 and G = Cir(n,A), where A= {1,2, . . . ,k,n−k, . . . ,n−1}. Suppose n=(2k+1)t+h for some positive integers t and h with1≤h≤2k and ℓ = ⌈ n
|A|+1⌉. Then D = {0,c+h,c+h +(2k+1), . . . ,c+h +(ℓ−2)(2k+1)}is aγ-set for G, where 1≤c≤2k−h+1.
Lemma 4.9([16, Lemma 3.2.14]). Let n(≥ 3), k be integers such that1 ≤ k ≤ n−1
2 and G = Cir(n,A), where A= {1,2, . . . ,k,n−k, . . . ,n−1}. If 2k+1divides n, then G is just excellent.
Lemma 4.10([16, Lemma 3.2.15]). Let n(≥3)be an even integer and k be such that1≤k≤ n−1
2 . Suppose D is a γ-set for G1=Cir(n,A1), where A1= {1, . . . ,k,n−k, . . . ,n−1}. Then D is a dominating set for G2=Cir(n,A2), where A2= {1, . . . ,k,n−k, . . . ,n−1,n2}.
The results given below identify independent and perfect domination numbers ofCir(n,A).
Theorem 4.11([11, Theorem 2.1]). Let n(≥ 3), k be integers such that1 ≤ k ≤ n−1
2 and 2k+1divides n. Let G=Cir(n,A), where A= {1,2, . . . ,k,n−k, . . . ,n−1}. Then i(G)=γp(G)= n
2k+1.
The above theorem gives a tool to identify an E- set inCir(n,A) and the same provided by the corollary given below.
Corollary 4.12([11, Corollary 2.2]). Let n(≥ 3), k be integers such that1 ≤ k ≤ n−1
2 and2k+1divides n. Let G=Cir(n,A), where A= {1,2, . . . ,k,n−k, . . . ,n−1}and D= {0,(2k+1),2(2k+1), . . . ,(⌈ n
2k+1⌉ −1)(2k+1)}. Then for each h with1≤h≤2k, we have D+h is both an independent and perfect dominating set of G.
Corollary 4.13([11, Corollary 2.3]). Let n(≥ 3), k be integers such that1 ≤ k ≤ n−1
2 and2k+1divides n. Let G=Cir(n,A), where A= {1,2, . . . ,k,n−k, . . . ,n−1}. Then di(G)=dp(G)=2k+1.
Remark 4.14([11, Remark 2.4]). For any vertexvofG=Cir(n,A),|N[v]| = |A| +1. In view ofLemma 4.5and Theorem 4.11,Ghas an efficient dominating set only when 2k+1 dividesn.
Corollary 4.15([16, Corollary 3.4.6]). Let n(≥3), k be integers such that1≤k ≤ n−1
2 and2k+1divides n. Let G=Cir(n,A), where A= {1,2, . . . ,k,n−k, . . . ,n−1}. Then G is domatically full.
Theorem 4.16([11, Theorem 2.5]). Let n(≥3), k be integers such that1≤k ≤ n−1
2 and2k+1does not divide n.
Let G=Cir(n,A), where A= {1,2, . . . ,k,n−k, . . . ,n−1}. Then i(G)= ⌈ n
2k+1⌉.
Theorem 4.17 ([11, Theorem 2.8]). Let n(≥ 3) and k be positive integers such that 1 ≤ k ≤ n−1
2 . Then i i(G)=γi(G)=γ γ(G)=2⌈ n
2k+1⌉, where G=Cir(n,A)and A= {1,2, . . . ,k,n−k, . . . ,n−1}. Lemma 4.18([16, Lemma 3.4.12]). Let n(≥ 3), k be integers such that1 ≤ k ≤ n−1
2 and G =Cir(n,A), where A= {1,2, . . . ,k,n−k, . . . ,n−1}. Then G isγ γ−minimum.
I. Rani [16] obtained the values of domination and independent domination numbers for the class of circulant graphs Cir(n,B), where B = {1,3, . . . ,2k −1,n −(2k −1), . . . ,n −3,n −1} with 1 ≤ k ≤ n−1
2 and
(2k−1)̸=n−(2k−1).
Theorem 4.19([16, Theorem 3.3.1]).Let n(≥3), k be integers such that1≤k≤ n−1
2 and(2k−1)̸=n−(2k−1). Let G=Cir(n,B), where B= {1,3, . . . ,2k−1,n−(2k−1), . . . ,n−3,n−1}. If 2k+1divides n, thenγ(G)= n
2k+1. Theorem 4.20([16, Theorem 3.4.13]). Let n(≥3)and k be integers such that1≤k≤ n−1
2 and(2k−1)̸=n−(2k−1).
Let G = Cir(n,B), where B = {1,3, . . . ,2k−1,n −(2k −1), . . . ,n −3,n −1}. If 2k+1 divides n, then i(G)=γp(G)= n
2k+1.
Note that the setDidentified inTheorem 4.20is anE-set inCir(n,B).
5. Connected domination in circulants
In this section, we present results concerning the total, connected and restrained domination parameters for Cir(n,A), whereA= {1,2, . . . ,k,n−k, . . . ,n−1}.
Theorem 5.1 ([12, Theorem 2.1]). Suppose n(≥ 3) and k are integers with 1 ≤ k ≤ n−1
2 . Assume that n = (ℓt − 1)(3k +1) +h for some h with 1 ≤ h ≤ k and ℓt = ⌈ n
3k+1⌉. Let G = Cir(n,A), where A= {1,2, . . . ,k,n−k, . . . ,n−1}. Thenγt(G)=2⌈3k+1n ⌉ −1.
Theorem 5.2([12, Theorem 2.2]). Let n(≥ 3), k be integers such that 1 ≤ k ≤ n−1
2 and 3k+1 divides n. Let G=Cir(n,A), where A= {1,2, . . . ,k,n−k, . . . ,n−1}. Thenγt(G)= 2n
3k+1.
Theorem 5.3 ([12, Theorem 2.3]). Suppose n(≥ 3) and k are integers with 1 ≤ k ≤ n−1
2 . Assume that n = (ℓt −1)(3k+1)+h for some h with k +1 ≤ h ≤ 3k and ℓt = ⌈ n
3k+1⌉. Let G = Cir(n,A), where A= {1,2, . . . ,k,n−k, . . . ,n−1}. Thenγt(G)=2⌈ n
3k+1⌉.
Theorem 5.4([12, Theorem 2.4]). Let n(≥3)be an even integer and k be an integer such that 1 ≤ k ≤ n−2
2 . Let G = Cir(n,A), where A = {1,2, . . . ,k,n−k, . . . ,n −1,n2}. If 4k+2 does not divide n, then2⌈ n
4k+2⌉ −1 ≤ γt(G)≤2⌈ n
4k+2⌉.
Lemma 5.5([12, Lemma 2.6]). Suppose n(≥ 3)and k are integers with1 ≤ k≤ n−1
2 . Let A1 = {1,n−1}, A2 = {1,2, . . . ,k,n−k, . . . ,n−1}. Then anyγt-set of G1=Cir(n,A1), is a total dominating set of G2=Cir(n,A2).
We see the connected domination number ofCir(n,A) in the following theorem.
Theorem 5.6([12, Theorem 3.1]). Suppose n(≥3)and k are integers with1≤k≤ n−1
2 . Let G=Cir(n,A), where A= {1,2, . . . ,k,n−k, . . . ,n−1}. Thenγc(G)= ⌈n−(2k+1)
k ⌉ +1.
Lemma 5.7([12, Lemma 3.2]). Let n(≥ 4)be an even integer and G =Cir(n,A), where A= {1,n−1,n2}. Then γc(G)≤ n
2 −1.
Theorem 5.8([12, Theorem 3.3]). Let n(≥4)be an even integer and k be an integer such that 1 ≤ k ≤ n−1
2 . Let G=Cir(n,A), where A= {1,2, . . . ,k,n−k, . . . ,n−1,n2}. Thenγc(G)≤2⌈n−2(2k+1)
2k ⌉ +2.
Lemma 5.9 ([12, Lemma 3.5]). Let n(≥ 4) be an even integer and k be an integer such that k = ⌈n−2
4 ⌉. Let G=Cir(n,A), where A= {1,2, . . . ,k,n−k, . . . ,n−1,n2}. Then T = {0,n2}is bothγt-set andγc-set of G.
We have the restrained domination number ofCir(n,A) in the following theorem.
Theorem 5.10([16, Theorem 4.4.1]). Suppose n,k are integers such that2 ≤k≤ n−1
2 and G =Cir(n,A), where A= {1,2, . . . ,k,n−k, . . . ,n−1}. Thenγr(G)= ⌈ n
2k+1⌉.
Theorem 5.11([16, Theorem 4.4.2]). Let n(≥3)be an integer and G =Cir(n,A), where A= {1,n−1}. Suppose n=3t or3t+1for some positive integer t . Thenγr(G)= ⌈n
3⌉.
Theorem 5.12([16, Theorem 4.4.3]). Let n(≥3)be an integer and G =Cir(n,A), where A= {1,n−1}. Suppose n=3t+2for some positive integer t . Thenγr(G)= ⌈n
3⌉ +1.
Theorem 5.13([16, Theorem 4.4.4]). Suppose n(≥3)and k are integers such that1≤k≤ n−1
2 and G=Cir(n,A), where A= {1,2, . . . ,k,n−k, . . . ,n−1}. Then1≤γr(G)≤ ⌈n
3⌉ +1.
Theorem 5.14([16, Theorem 4.4.5]). Let n(≥3)be an even integer and k be an integer such that1≤k≤ n−1
2 . Let G1=Cir(n,A1), where A1 = {1,2, . . . ,k,n−k, . . . ,n−1}and G2 =Cir(n,A2), where A2 = {1,2, . . . ,k,n− k, . . . ,n−1,n2}. Anyγr-set of G1is a restrained dominating set of G2.
6. Domination in another class of circulants
In the previous section, we have presented results on domination parameters for the circulant graphsCir(n,A), where A = {1,2, . . . ,k,n −k, . . . ,n −1} with 1 ≤ k ≤ n
2. This section focuses on the domination number, total domination number and connected domination number for circulant graphs with respect to the generating sets {m−(k−1), . . . ,m−1,m,n−m,n−(m−1), . . . ,n−(m−(k−1))}whennis odd and{m−(k−1), . . . ,m− 1,m,n2,n−m,n−(m−1), . . . ,n−(m−(k−1))}whennis even. Here,mandkare integers such thatm= ⌊n−1
2 ⌋ and 1≤k≤m.
Lemma 6.1([17, Lemma 3.2.1]). Let n(≥3)be an odd integer, m= n−1
2 and k be an integer such that1≤k≤m.
Let G=Cir(n,A), where A= {m−(k−1), . . . ,m−1,m,n−m,n−(m−1), . . . ,n−(m−(k−1))}. If⌈ n
2k+1⌉ is odd, thenγ(G)= ⌈ n
2k+1⌉.
Corollary 6.2([17, Corollary 3.2.2]). Let n(≥3)be an odd integer, m = n−1
2 and k be an integer such that2k+1 divides n with1≤k≤m. Let G=Cir(n,A), where A= {m−(k−1), . . . ,m−1,m,n−m,n−(m−1), . . . ,n− (m−(k−1))}. Then G has an efficient dominating set.
Whennand⌈ n
2k+1⌉are odd, it is proved thatγ(G)= ⌈ n
2k+1⌉. But it is not true in general when⌈ n
2k+1⌉is even. The next lemma gives the value of the domination number when⌈ n
2k+1⌉is even.
Lemma 6.3([17, Lemma 3.2.4]). Let n(≥3)be an odd integer, m= n−1
2 and k be an integer such that1≤k≤m.
Let G=Cir(n,A), where A= {m−(k−1), . . . ,m−1,m,n−m,n−(m−1), . . . ,n−(m−(k−1))}. (a) If ℓ= ⌈ n
2k+1⌉is even and m−k=(2ℓ−1)(2k+1)+1, thenγ(G)= ⌈ n
2k+1⌉. (b) Ifℓ= ⌈ n
2k+1⌉is even and m−k=(ℓ2−1)(2k+1)+j , for some integer j with2≤ j ≤k, thenγ(G)≤ ⌈ n
2k+1⌉+1.
UsingLemmas 6.1and6.3, the following theorem is proved which gives lower and upper bounds for domination number.
Theorem 6.4([17, Theorem 3.2.5]). Let n(≥3)be an odd integer, m= n−1
2 and k be an integer such that1≤k≤m.
Let G = Cir(n,A), where A = {m−(k−1), . . . ,m−1,m,n −m,n−(m−1), . . . ,n −(m−(k−1))}. Then
⌈ n
2k+1⌉ ≤γ(G)≤ ⌈ n
2k+1⌉ +1.
Lemma 6.5([17, Lemma 3.2.6]). Let n(≥3)be an even integer, m= ⌊n−1
2 ⌋and k be an integer such that1≤k≤m.
Let G=Cir(n,A), where A= {m−(k−1), . . . ,m−1,m,n2,n−m,n−(m−1), . . . ,n−(m−(k−1))}. If ⌈ n
2k+2⌉ is odd, thenγ(G)= ⌈ n
2k+2⌉.
Corollary 6.6([17, Corollary 3.2.7]). Let n(≥3)be an even integer, m = ⌊n−1
2 ⌋and k be an integer such that2k+2 divides n with1≤k≤m. Let G=Cir(n,A), where A= {m−(k−1), . . . ,m−1,m,n2,n−m,n−(m−1), . . . ,n− (m−(k−1))}. If 2k+2n is odd, then G has an efficient dominating set.
Lemma 6.7([17, Lemma 3.2.8]). Let n(≥3)be an even integer, m= ⌊n−1
2 ⌋and k be an integer such that1≤k≤m.
Let G=Cir(n,A), where A= {m−(k−1), . . . ,m−1,m,n2,n−m,n−(m−1), . . . ,n−(m−(k−1))}. (a) If ℓ= ⌈ n
2k+2⌉is even and m−k=(2ℓ−1)(2k+2)+1, thenγ(G)= ⌈ n
2k+2⌉. (b) If ℓ = ⌈ n
2k+2⌉is even and m−k = (ℓ2 −1)(2k+2)+ j , for some integer j with 2 ≤ j ≤ k+1, then
γ(G)≤ ⌈ n
2k+2⌉ +1.
UsingLemmas 6.5and6.7, the following result is arrived.
Theorem 6.8([17, Theorem 3.2.9]). Let n(≥ 3) be an even integer, m = ⌊n−1
2 ⌋ and k be an integer such that 1≤k≤m. Let G=Cir(n,A), where A= {m−(k−1), . . . ,m−1,m,n2,n−m,n−(m−1), . . . ,n−(m−(k−1))}. Then⌈ n
2k+2⌉ ≤γ(G)≤ ⌈ n
2k+2⌉ +1.
7. Total and efficient open domination
In this section, we list the results on the total domination number and a corresponding minimum total dominating set for the class of circulant graphsCir(n,A), whereA= {m−(k−1), . . . ,m−1,m,n−m,n−(m−1), . . . ,n− (m−(k−1))}whennis odd and A= {m−(k−1), . . . ,m−1,m,n2,n−m,n−(m−1), . . . ,n−(m−(k−1))} whennis even. Note that certain circulant graphs in this class admit efficient open dominating sets.
Lemma 7.1([18, Lemma 2.1]). Let n(≥3)be an odd integer, m = n−1
2 and k be an integer such that1 ≤k ≤m.
Let G = Cir(n,A), where A = {m−(k−1), . . . ,m−1,m,n −m,n−(m−1), . . . ,n −(m−(k−1))}. Then γt(G)= ⌈n
2k⌉.
Corollary 7.2([17, Corollary 3.3.2]). Let n(≥3)be an odd integer, m= n−1
2 and k be an integer such that2k divides n with1≤k≤m. Let G=Cir(n,A), where A= {m−(k−1), . . . ,m−1,m,n−m,n−(m−1), . . . ,n−(m−(k−1))}. Then G has an efficient open dominating set.
Lemma 7.3([18, Lemma 2.2]). Let n(≥3)be an even integer, m= ⌊n−1
2 ⌋and k be an integer such that1≤k≤m.
Let G =Cir(n,A), where A= {m−(k−1), . . . ,m−1,m,n2,n−m,n−(m−1), . . . ,n−(m−(k−1))}. Then γt(G)= ⌈ n
2k+1⌉.
Corollary 7.4([17, Corollary 3.3.5]). Let n(≥3)be an even integer, m = ⌊n−1
2 ⌋and k be an integer such that2k+1 divides n with1≤k≤m. Let G=Cir(n,A), where A= {m−(k−1), . . . ,m−1,m,n2,n−m,n−(m−1), . . . ,n− (m−(k−1))}. Then G has an efficient open dominating set.
InLemma 7.3, each interval (exceptIℓ) contains exactly 2k+1 vertices and 1≤ |Iℓ| ≤2k+1.From this, one can find that the vertices 1,2, . . . ,(2k+1)−j are dominated by bothx(=m+k+2) andx+(ℓ−1)(2k+1) and they are the only vertices dominated by two vertices in theγt-setDtspecified inLemma 7.3.
Since the vertices of V(G) are group elements, Dt +y is a γt-set for all y ∈ V(G). This implies that G is total excellent. In particular, D′t = Dt + (n −x) = {0,2k,2(2k), . . . ,(ℓ−1)2k} when n is odd and D′′t = Dt+(n−x)= {0,(2k+1),2(2k+1), . . . ,(ℓ−1)(2k+1)}whennis even, are alsoγt-sets ofG=Cir(n,A) with respective to the generating setsA= {m−(k−1), . . . ,m−1,m,n−m,n−(m−1), . . . ,n−(m−(k−1))}and A= {m−(k−1), . . . ,m−1,m,n2,n−m,n−(m−1), . . . ,n−(m−(k−1))}, respectively. Further, certain 2-total excellent circulant graphs are identified in the following results.
Lemma 7.5([18, Lemma 3.1]). Let n(≥3)be an odd integer, m= n−1
2 and k be an integer such that1≤k≤m. Let G=Cir(n,A), where A= {m−(k−1), . . . ,m−1,m,n−m,n−(m−1), . . . ,n−(m−(k−1))}. If n=t(2k)+1 for some integer t >0, then G is 2-total excellent.
Theorem 7.6([18, Theorem 3.2]). Let n(≥3)be an odd integer, m= n−1
2 and k be an integer such that1≤k≤m.
Let G=Cir(n,A), where A= {m−(k−1), . . . ,m−1,m,n−m,n−(m−1), . . . ,n−(m−(k−1))}. If n=t(2k)+j for some integers t(>0)and j with1≤ j ≤2k, then G is 2-total excellent if and only if j=1.
Lemma 7.7([18, Lemma 3.3]). Let n(≥3)be an even integer, m= ⌊n−1
2 ⌋and k be an integer such that1≤k≤m.
Let G = Cir(n,A), where A = {m−(k−1), . . . ,m−1,m,n2,n−m,n−(m−1), . . . ,n −(m−(k−1))}. If n=t(2k+1)+1for some integer t(>0), then G is 2-total excellent.
Theorem 7.8 ([18, Theorem 3.4])). Let n(≥ 3) be an even integer, m = ⌊n−1
2 ⌋and k be an integer such that 1≤k≤m. Let G=Cir(n,A), where A= {m−(k−1), . . . ,m−1,m,n2,n−m,n−(m−1), . . . ,n−(m−(k−1))}. If n=t(2k+1)+ j for some integers t(>0)and j with1 ≤ j ≤2k+1, then G is 2-total excellent if and only if
j =1.
8. Domination in general circulants
In this section, we concentrate on upper bounds for the domination number, total domination number and connected domination number of an arbitrary circulant graph. Further, it is proved that the upper bound is reachable in certain cases. In those cases, the domatic number and independent domatic number are also obtained. Throughout this section, the generating set A of Zn is taken as A = {a1,a2, . . . ,ak,n −ak,n −ak−1, . . . ,n −a1}, where 1≤a1<a2<· · ·<ak ≤m,d1=a1,di =ai−ai−1for 2≤i ≤kandd = max
1≤i≤k{di}. Lemma 8.1([14, Lemma 2.1]). Let n(≥3)be an integer, m= ⌊n−1
2 ⌋and k be an integer such that1≤k≤m. Let G=Cir(n,A), where A= {a1,a2, . . . ,ak,n−ak,n−ak−1, . . . ,n−a1}. If d1 =a1, di =ai −ai−1for2≤i ≤k and d= max1≤i≤k{di}, thenγ(G)≤d⌈ n
d+2ak⌉.
By takingd1=d2= · · · =dk=d, the following corollary is proved.
Corollary 8.2 ([17, Corollary 4.1.2]). Let n(≥ 3) be an integer, m = ⌊n−1
2 ⌋ and k,d be integers such that 1 ≤ kd ≤ m. Let G = Cir(n,A), where A = {d,2d, . . . ,kd,n − kd,n − (k − 1)d, . . . ,n − d}. Then
γ(G)≤d⌈ n
d(1+2k)⌉.
Theorem 8.3([14, Theorem 2.3])). Let n(≥3)be an integer, m= ⌊n−1
2 ⌋and k,d be integers such that1≤kd≤m.
Let G = Cir(n,A), where A = {d,2d, . . . ,kd,n −kd,n −(k−1)d, . . . ,n−d}. If d(1+2k)divides n, then
γ(G)= n
1+2k. In this case, G has an efficient dominating set.
Lemma 8.4([14, Lemma 2.4]). Let n(≥ 3)be an integer, m = ⌊n−1
2 ⌋and k,d be integers such that1 ≤kd ≤m.
Let G=Cir(n,A), where A= {d,2d, . . . ,kd,n−kd,n−(k−1)d, . . . ,n−d}. Suppose d(1+2k)divides n, then d(G)=di(G)=dp(G)=2k+1.
Lemma 8.5([14, Lemma 2.2]). Let n(≥ 3) be an integer, k,d be integers such that 1 ≤ kd ≤ m = ⌊n−1
2 ⌋ and ℓ = ⌈ n
d+2kd⌉. Let G = Cir(n,A), where A = {d,2d, . . . ,kd,n−kd,n −(k −1)d, . . . ,n−d}. Suppose n=ℓ(d+2kd)+j for some integer j with1≤ j ≤d(1+2k). Then
(a) γ(G)≤d(ℓ−1)+j if 1≤ j ≤d−1.
(b) γ(G)≤dℓotherwise.
The results given below identify an upper bound for the total domination number and the connected domination number for a general circulant graph.
Theorem 8.6([14, Theorem 3.1]). Let n(≥3)be an integer, m = ⌊n−1
2 ⌋and k be an integer such that1≤k≤m. Let G=Cir(n,A), where A= {a1,a2, . . . ,ak,n−ak,n−ak−1, . . . ,n−a1}. If d1 =a1, di =ai −ai−1for2≤i ≤k and d= max1≤i≤k{di}, thenγt(G)≤2d⌈ n
d+3ak⌉.
Corollary 8.7 ([17, Corollary 4.2.2]). Let n(≥ 3) be an integer, m = ⌊n−1
2 ⌋ and k,d be integers such that 1 ≤ kd ≤ m. Let G = Cir(n,A), where A = {d,2d, . . . ,kd,n − kd,n − (k − 1)d, . . . ,n − d}. Then γt(G)≤2d⌈ n
d(1+3k)⌉.
In the above corollary, ifd=1, then we haveγt(G)≤2⌈ n
1+3k⌉. Since∆(G)=2kand from the nature of elements inA, any two adjacent vertices can dominate at most 3k+1 different vertices ofG. Thus,γt(G) ≥2⌈ n
1+3k⌉and so γt(G)=2⌈ n
1+3k⌉.
Lemma 8.8([14, Lemma 3.2]). Let(n ≥3)be an integer, m = ⌊n−1
2 ⌋and k,d be integers such that1≤ kd ≤m.
Let G=Cir(n,A), where A= {d,2d, . . . ,kd,n−kd,n−(k−1)d, . . . ,n−d}. Suppose n=ℓ(d+3kd)+ j for some1≤ j≤d+3kd, whereℓ= ⌈ n
d+3kd⌉, then (a) γt(G)≤2d(ℓ−1)+ j , if 1≤ j≤d−1;
(b) γt(G)≤2dℓ, otherwise.
Lemma 8.9([14, Lemma 3.3]). Let(n ≥3)be an integer, m = ⌊n−1
2 ⌋and k,d be integers such that1≤ kd ≤m.
Let G=Cir(n,A), where A= {d,2d, . . . ,kd,n−kd,n−(k−1)d, . . . ,n−d}. Suppose d(1+3k)divides n, then γt(G)= 2n
1+3k.
Lemma 8.10([14, Lemma 3.5]). Let n(≥ 3)be an integer, m = ⌊n−1
2 ⌋and k be an integer such that 1 ≤k ≤ m.
Let G=Cir(n,A), where A= {1,a2, . . . ,ak,n−ak,n−ak−1, . . . ,n−a2,n−1}. If d1 =1, di =ai−ai−1for 2≤i≤k and d= max1≤i≤k{di}, thenγc(G)≤d(1+ ⌈n−(d+2ak)
ak ⌉).
9. Efficient dominating sets in circulant graphs of large degree
In this section, results about wreath products of circulant graphs are presented and they identify infinitely many circulant graphs with efficient dominating sets whose elements need not be equally spaced in Zn. Kumar and MacGillivray [19] proved that if a circulant graph of large degree has an efficient dominating set, then either its elements are equally spaced or the graph is the wreath product of a smaller circulant graph with an efficient dominating set and a complete graph.
The following statement about wreath products of circulant graphs was proved by Broere and Hattingh [20]. For d ∈ Zn, the symbol⟨d⟩denotes the subgroup ofZn generated byd. The wreath product of two graphs GandH is denoted byG∼H.
Theorem 9.1([20, Proposition 27]). If G =Cir(n,S)and H =Cir(m,T)are circulant graphs, then G ∼ H is isomorphic to F=Cir(mn,nT∪(⋃
s∈S
nZm+s)). Further, the subgraph of F induced by choosing one representative from each coset of ⟨n⟩is isomorphic to G, and the subgraph of F induced by any coset of⟨n⟩is isomorphic to H .
Proposition 9.2([19, Proposition 3.2]). Let G be a graph without isolated vertices and H be a graph with at least one vertex. Then G∼H has an efficient dominating set if and only if G has an efficient dominating set andγ(H)=1.
Theorem 9.3([19, Theorem 3.3]). The graph G =Cir(mn,S)is isomorphic to the wreath product of a circulant graph on n vertices and Kmif and only if S∪ {0}is a union of cosets of ⟨n⟩.
Corollary 9.4([19, Corollary 3.4]). Let F =Cir(mn,T)and suppose T ∪ {0}is a union of cosets of ⟨n⟩. Let S be any set of representatives of the cosets⟨n⟩ +t , t ∈T − ⟨n⟩. Then F has an efficient dominating set if and only if G=Cir(n,T)has an efficient dominating set.
The largest possible values of the degree of ann-vertex circulant graph, which is not complete and which might have an efficient dominating set, aren2−1 andn3−1. These graphs have domination number two and three, respectively.
The above ideas are used to completely describe their efficient dominating sets.
We call a circulant graph reducible if it is a wreath product of a smaller circulant graph and a complete graph with at least two vertices; otherwise, we call it irreducible. ByTheorem 9.3,Cir(n,S) is reducible if and only ifS∪ {0}is a union of cosets of⟨d⟩for some divisordofnwithd>1.
Proposition 9.5 ([19, Proposition 4.1]). Suppose that the reducible circulant graph Cir(n,S) has an efficient dominating set. If d is the smallest positive integer such that S∪ {0}is a union of cosets of ⟨d⟩, then d divides n, m=n/d divides|S| +1and Cir(n,S)is the wreath product of an irreducible circulant graph and Km.
Theorem 9.6([19, Theorem 4.2]). Suppose|S| = k−1and G = Cir(2k,S)has an efficient dominating set, D.
Then either S∪ {0}is a union of cosets of a nontrivial proper subgroup of Z2k, or D is a coset of ⟨k⟩.
Corollary 9.7([19, Corollary 4.3]). Suppose that G =Cir(2k,S), where|S| =k−1, has an efficient dominating set.
(a) If G is irreducible, then every efficient dominating set is equally spaced. In particular,{0,k}is an efficient dominating set.
(b) If G is reducible, then there exist integers n and m such that m≥2and G is the wreath product of the irreducible circulant graph Cir(2n,T), where|T| =n−1, and Km. Further, Cir(2n,T)has an efficient dominating set.
Theorem 9.8([19, Theorem 4.4]). Suppose|S| = k−1and G = Cir(3k,S)has an efficient dominating set, D.
Then either S∪ {0}is a union of cosets of a nontrivial proper subgroup of Z3k, or D is a coset of ⟨k⟩.
Corollary 9.9([19, Corollary 4.5]). Suppose that G =Cir(3k,S), where|S| =k−1, has an efficient dominating set.
(a) If G is irreducible, then every efficient dominating set is equally spaced. In particular,{0,k,2k}is an efficient dominating set.
(b) If G is reducible, then there exist integers n and m such that m≥2and G is the wreath product of the irreducible circulant graph Cir(3n,T), where|T| =n−1, and Km. Further, Cir(3n,T)has an efficient dominating set.
10. Subgroups as efficient dominating sets
As circulant graphs are constructed from the groupZn, there is a relationship between the substructures of the groupZn and that of the corresponding graphCir(n,A). Trivially, every efficient dominating set ofCir(n,A) is a subset of the groupZn and so there is a possibility for an efficient dominating set ofCir(n,A) to be a subgroup of Zn. With this in mind, a necessary and sufficient condition was obtained by Tamizh Chelvam and Mutharasu [15] for a subgroup ofZnto be an efficient dominating set ofCir(n,A) for some suitable generating setAofZn.
Lemma 10.1([15, Lemma 3.1]). Let H be a proper subgroup of Zn such that n = |H|(2k+1) for some integer k≥ 1. Then there exists a generating set A of Zn such that H is an efficient dominating set for the circulant graph G=Cir(n,A).
Corollary 10.2([15, Corollary 3.2]). Let H be a proper subgroup ofZnsuch that n= |H|(2k+1)for some integer k≥1. Then there exists a generating set A of Znsuch that for every x ∈Γ, the co-set H+x is an efficient dominating set for the circulant graph G=Cir(n,A).
Lemma 10.3([15, Lemma 3.3]). Let H be a proper subgroup of Zn such that|H|is odd and n = |H|(2k+2)for some integer k ≥ 1. Then there exists a generating set A of Zn such that H is an efficient dominating set for the circulant graph G=Cir(n,A).
By the above lemma and the property of vertex transitivity on circulant graphs, the following corollary was obtained.
Corollary 10.4([15, Corollary 3.4]). Let H be a proper subgroup ofZn such that|H|is odd and n= |H|(2k+2) for some integer k≥1. Then there exists a generating set A of Znsuch that for every x ∈Γ, the co-set H+x is an efficient dominating set for the circulant graph G=Cir(n,A).
Theorem 10.5([15, Theorem 3.6]). Let H be a proper subgroup of Zn. Then H (as well as H +x for any x ∈Zn) is an efficient dominating set in G =Cir(n,A)for some suitable generating set A of Zn if and only if either of the following is true.
(a) n= |H|(2k+1)for some integer k≥1;
(b) n= |H|(2k+2)for some integer k≥1and|H|is odd.
Acknowledgments
This research work is supported by the SERB Project No. SR/S4/MS:806/13 of Science and Engineering Research Board, Government of India for the first author. Also it is supported through the INSPIRE programme ( IF160672) of Department of Science and Technology, Government of India for the second author.
References
[1] T.W. Haynes, S.T. Hedetniemi, P.J. Slater, Domination in Graphs Advanced Topics, Marcel Decker, New York, 1998.
[2] I.J. Dejter, O. Serra, Efficient dominating sets in Cayley graphs, Discrete Appl. Math. 129 (2003) 319–328.
[3] Hamed Hatami, Pooya Hatami, Perfect dominating sets in the Cartesian products of prime cycles, Electron. J. Combin. 14 (2007) # N8.
[4] Jia Huang, Jun-Ming Xu, The bondage numbers and efficient domination of vertex-transitive graphs, Discrete Math. 308 (2008) 571–582.
[5] Italo J. Dejter, Perfect domination in regular grid graphs, Australas. J. Combin. 42 (2008) 99–114.
[6] J. Lee, Independent perfect domination sets in Cayley graphs, J. Graph Theory 37 (4) (2001) 213–219.
[7] N. Obradovi´c, J. Peters, Goran Ru˘zi´c, Efficient domination in circulant graphs with two chord lengths, Inform. Process. Lett. 102 (2007) 253–258.
[8] T. Tamizh Chelvam, M.S. Mutharasu, Efficient open domination in Cayley graphs, Appl. Math. Lett. 25 (2012) 1560–1564.http://dx.doi.org/
10.1016/j.aml.2011.12.036.
[9] Young Soo Kwon, Jaeun Lee, Perfect domination sets in Cayley graphs, Discrete Appl. Math. 162 (2014) 259–263.http://dx.doi.org/10.1016/
j.dam.2013.09.020.
[10] T. Tamizh Chelvam, I. Rani, Dominating sets in Cayley Graphs onZn, Tamkang J. Math. 38 (4) (2007) 341–345.
[11] T. Tamizh Chelvam, I. Rani, Independent domination number of Cayley graphs onZn, J. Combin. Math. Combin. Comput. 69 (2009) 251–255.
[12] T. Tamizh Chelvam, I. Rani, Total and Connected Domination numbers of Cayley graphs onZn, Adv. Stud. Contemp. Math.(Kyungshang) 20 (1) (2010) 57–61.
[13] T. Tamizh Chelvam, M.S. Mutharasu, I. Rani, Domination in Generalized Circulant Graphs, J. Indian Math. Soc. 52 (3) (2010) 531–541.
[14] T. Tamizh Chelvam, M.S. Mutharasu, Bounds for Domination Parameters in Circulant Graphs, Adv. Stud. Contemp. Math. (Kyungshang) 22 (4) (2012) 525–529.
[15] T. Tamizh Chelvam, M.S. Mutharasu, Subgroups as efficient dominating sets in Cayley graphs, Discrete Appl. Math. 161 (8) (2013) 1187–1190.http://dx.doi.org/10.1016/j.dam.2012.09.005.
[16] I. Rani, Domination in Circulant Graphs (Ph.D dissertation), Manonmaniam Sundaranar University, 2010.
[17] M.S. Mutharasu, Domination in Cayley Graphs (Ph.D dissertation), Manonmaniam Sundaranar University, 2011.
[18] T. Tamizh Chelvam, M.S. Mutharasu, Total Domination in Circulant Graphs, Int. J. Open Probl. Compt. Math. 4 (2) (2011) 168–174.
[19] K. RejiKumar, G. MacGillivray, Efficient domination in circulant graphs, Discrete Math. 313 (2013) 767–771.
[20] I. Broere, J. Hattingh, Products of circulant graphs, Quaest. Math. 13 (1990) 191–216.
