Chapter 5 Complete Cototal Domination Number of a Graph Published in Journal of Scientific Research Vol. () (2011), 547-555 (Bangladesh). 64
ABSTRACT Let G = (V,E) be a graph. A dominating set D V is said to be complete cototal dominating set if every vertex in V is adjacent to some vertex in D and there exists a vertex u D and v V D such that uv E(G) and N(v) u = x V D. The completecototaldomination number γ cc (G) of G is the minimum cardinality of a complete cototal dominating set of G. In this chapter, we initiate the study of complete cototal domination in graphs and present bounds and some exact values for γ cc (G). Also its relationship with other domination parameters are established and related two open problems are explored. 65
5.1 Introduction The graphs considered in this chapter are simple and without isolated vertices. A vertex v V is called a pendant vertex, if deg G (v) = 1 and an isolated vertex if deg G (v) = 0, where d(g) = deg G (x) is the degree of a vertex x V(G). A vertex which is adjacent to a pendant vertex is called a support vertex. A dominating set D V of a graph G = (V,E) is called a connected dominating set if the induced subgraph D is connected. The connected domination number γ c (G) of G is the minimum cardinality of a connected dominating set of G [59]. A set D V of a graph G = (V,E) is called a total dominating set if the induced subgraph D has no isolated vertices. The total domination number γ t (G) of G is the minimum cardinality of a total dominating set of G [22]. A dominatingset D issaidtobe a cototaldominatingset if the induced subgraph V D has no isolated vertices. The cototal domination number γ cl (G) of G is the minimum cardinalityof a cototaldominating set of G [44]. 66
Complete Cototal Domination Number of a Graph Inspired from the cototal domination, we introduce the new parameter in the field of domination theory of graphs as follows. Definition 5.1. Let G = (V,E) be a graph. A dominatingset D V is saidtobe completecototaldominatingset ifeveryvertexin V isadjacent to some vertex in D and there exists a vertex u D and v V D such that uv E(G) and N(v) u = x V D. The complete cototal domination number γ cc (G) of G is the minimum cardinality of a complete cototal dominating set of G. For illustration, consider a graph in Figure 5.1. 1 G : 5 6 4 2 Figure 5.1 In Figure 5.1, the set {1,2,} is a dominatingset as well as cototal dominating set, and set {4,5,6} is a total dominating set. However, any com- 67
plete cototal dominating set must include each pendant vertex {1, 2, } and its neighbor {4,5,6}. Thus γ cc (G) = {1,2,,4,5,6}= V = p. Applications Many applications of domination in graphs can be extended to complete cototal domination. For example, if we think of each vertex in a dominating set as file server for a computer network, then each computer in the network has direct access to a file server. It is sometimes reasonable to assume that this access be available even when one of the file servers goes down. A complete cototal dominating set provides the desired fault tolerance for such cases because each computer has access to at least two file servers and each file server has direct access to one backup server and each backup file server has direct access to at least one backup file server. 5.2 Results Obviously, a graph with an isolated vertex cannot have a complete cototal dominating set. We ask the natural question regarding the existence of complete cototal dominating sets. 68
Theorem 5.1 Every graph with no isolated vertices has a complete cototal dominating set and hence a complete cototal domination number. Proof. Without loss of generality, let G = (V,E) be connected. Then V itself is a complete cototal dominating set as each vertex is considered to dominateitself,andsince G hasnoisolates, G isnontrivialsoeachvertex v isadjacenttosomeothervertex u. Thusboth u and v dominate v and if V D has no isolated vertices. Now that we know G has a complete cototaldominatingset, wemayremoveonevertexatatimefrom V ifand only if the remaining subset of V is still a complete cototal dominating set. This will give a minimal complete cototal dominating set. Among all the complete cototal dominating sets, each of the smallest sets has cardinality γ cc (G). Henceforth we consider only graphs with no isolated vertices. First, we calculate complete cototal domination in specific families of graphs. Theorem 5.2 (i) For any path P p, γ cc (P p ) = p 2 p 2 4. (ii) For any cycle C p, γ cc (C p ) = p 2 p 4. (iii) For any star K 1,p 1, γ cc (K 1,p 1 ) = p. 69
(iv) For any wheel W p, γ cc (W p ) = 2. (v) For any complete graph K p, γ cc (K p ) = p if p = 2, 2 otherwise. Next, we state straightforward upper and lower bounds for γ cc (G). Theorem 5. Let G be a graph with no isolated vertices. Then 2 γ cc (G) p, and these bounds are sharp. Further, equality of an upper bound holds if and only if every edge of G is incident to a support vertex or G = C. Proof. Clearly from the definition of γ cc (G) and by Theorem 5.2, the result follows. Now for the equality of an upper bound, if p = 2 or then the result is obvious. Let us assume that p 4. Let D be the complete cototal dominating set of G. Then clearly D contains all the pendant vertices and support vertices of a graph G. Also no two vertices of V D are adjacent, which implies that D = p is a minimum complete cototal dominating set of G. Suppose every edge of G is not incident to a support vertex or G = C, 70
then there exists an edge uv where neither u nor v is a support vertex and deg(u) 2 and deg(v) 2. Let D = V {u,v}. Obviously D is a dominating set and V D is connected. If D contains no isolated vertices, then D is a complete cototal dominating set of G of cardinality p 2, a contradiction. Theorem 5.4 Let G = T be any nontrivial tree. Then 1+Δ(G) γ cc (G). Further equality holds if and only if G = K 1,p 1. Proof. Let G = T beanynontrivialtreeandlet D be acompletecototal dominating set of T. Clearly D contains all the pendant vertices and support vertices of a tree. Therefore, γ cc (T) 1+Δ(T). For equality, if T = K 1,p 1 then by result (iii) of Theorem 5.2, the result follows. Conversely, let T be a tree with γ cc (T) = 1 + Δ(T). Let D be a complete cototal dominating set with size D = 1+Δ(T) and let v be a maximum degree vertex in T. If v / D, then D 2Δ(T) which is impossible,so v D. Nowitfollowsfrom D = 1+Δ(T) that N(v) T and any vertex of N(v) is a pendant vertex. Hence D = K 1,p 1. Next theorem gives the upper bound for γ cc (G) in terms of order and 71
size. Theorem 5.5 Let G = (p, q) be any nontrivial connected graph. Then γ cc (G) 2q p+2. Further, equality holds if and only if G = T is tree and every edge of a tree is incident with a support vertex. Proof. Obviously, γ cc (G) p = 2(p 1) p+2 andsince G isconnected, q p 1. Thus γ cc (G) 2q p+2 if and only if every edge of a tree is incidentwithasupportvertex,then q = p 1 and γ cc (G) = p = 2q p+2. Conversely, let γ cc (G) = 2q p+2. Then 2q p+2 p, which implies that q p 1, so G is a tree with γ cc (G) = p and by Theorem 5., the result follows. Next, we give the upper bound for γ cc (G) in terms of order and minimum degree. Theorem 5.6 Let G be any connected graph of order at least four and 2 δ(g) p 1. Then γ cc (G) p δ(g). The next theorem relates γ cc (G) with maximum number of independent edges β 1 (G) of G. Theorem 5.7 If G has δ(g) 2, then γ cc (G) 2β 1 (G). 72
Proof. Let G be a graph with δ(g) 2 and M be a maximum independent set of edges in G. Let S M be the total dominating set of G. Since each v V S must have at least one neighbor in V S. So S is a complete cototal dominating set of G. Hence γ cc (G) 2β 1 (G). Theorem 5.A[27]. For any graph G without isolated vertices, γ(g) p Δ(G)+1. Relates the domination number to the maximum degree. We establish a similar result for γ cc (G). Theorem 5.8 For any graph G, without isolated vertices, 2p Δ(G)+1 γ cc(g) Proof. Let G have no isolated vertices and let S be a γ cc set for G. Let t denote the number of edges in G having one vertex in S and the other vertex in V S. Since Δ(G) deg(v) for all v S and each vertex in S is adjacent to at least one member of S, we have, t (Δ(G) 1) S = (Δ(G) 1)γ cc (G). Also each vertex in V S is adjacent to at least one vertex of V S, we have, t 2 V S = 2[p γ cc (G)]. 7
Hance, 2(p γ cc (G)) Δ(G)γ cc (G) γ cc (G). Whichreducestothebound of the theorem. Theorem 5.9 For any tree T = K 1,p 1 with with p 6, then γ cc (T) 2p. Next, we constructively characterize the tree for which γ cc (T) = 2p. We define the following operations on trees which achieve the lower bound of the Theorem 5.9. Operation 5.1. Attach P to a vertex v, where v is neither a pendant vertex nor a support vertex and N(v) has a support vertex of degree two in T. Operation 5.2. Attach P 1 to a vertex v, where v is either a pendant vertex or a support vertex of T. Operation 5.. Attach P 2 to any vertex of T. Let, X 0 = {T/T is a tree obtained from P 6 by finite type of operations 5.1}. X 1 = {T/Tis obtained from a tree T X 0 by applying operation 5.2 to T }. X 2 = {T/T is a tree which can be obtained from a tree T X by applying operation 5.}. Let, τ n = {T/Tis a tree of order p such that γ cc (T) = 2p }. 74
Proposition 5.1. If T X 0 has order p 6, then D(T) = P(T) S(T) is a complete cototal dominating set of cardinality 2p. Where D(T) is the union of set of all pendant vertices P(T) and the support vertices S(T) of a tree T. Proof. We use induction on k, the number of steps required to construct T. If k = 0, then T = P 6, and D(T) is a complete cototal dominating set of cardinality 4, i.e. 2p = 2 6 = 4. Suppose T can be constructed from P 6 by a sequence of k 1 operations 5.1, and suppose that T is a tree which can be obtained from P 6 by k 1 operations of type 1, then let D(T ) is a complete cototal dominating set of cardinality 2p(T ). As T X 0, T can be obtained from T by applying operation type 1. By induction assumption D(T ) = P(T ) S(T ) is a complete cototal dominating set of cardinality 2p(T ). Therefore, D(T ) {v 1,v 2 } = D(T) is a complete cototal dominating set of cardinality 2p(T). Proposition 5.2. Let T τ and v is a pendant vertex of T. If D {v} is a complete cototaldominatingset of T v, then the number of vertices in a tree is not a multiple of and T v τ. Proof. Suppose v is a pendant vertex of a tree T, and D {v} is a complete cototal dominating set of T {v}. Note that 75
2(p 1) γ cc (T v) 2p 1. If the number of vertices in a tree T is a multiple of three, i.e. p = k for some positive integer k 1, then we have 2(k 1) 2(k) 1, and so, 2k 2k 1, which is impossible. Thus p = k+1 or p = k+2 for some positive integer k 1. If p = k + 1, then 2(k) γ cc (T v) 2(k+1) 1, which implies that, 2k γ cc (T v) 2k and so T v τ. If p = k +2, then 2(k+1) γ cc (T v) 2(k+2) 1, which implies 2k +1 γ cc (T v) 2k +1 and so T v τ. Proposition 5.. Let T τ and suppose v 1 and v 2 are either both pendant vertices of T or v 1 is degree two support vertex of T adjacent to a pendant vertex v 2. If D {v 1,v 2 } is a complete cototal dominating set of T v 1 v 2, then T v 1 v 2 τ and the number of vertices in a tree minus two is a multiple of three. Proof. Suppose v 1 and v 2 are either pendant vertices of a tree T or v 1 is a degree two support vertex of T adjacent to a pendant vertex v 2 and that D {v 1,v 2 } is a complete cototal dominating set of T v 1 v 2. Note that 76
2(p 2) γ cc (T v 1 v 2 ) 2p 2. Suppose either p = k or p = k +1 for some positive integer k 1. If p = k, then 2(k 2) 2(k) 2, and so 2k 1 2k 2, which is impossible. If p = k+1, then 2(k 1) 2(k+1) 2, and so, 2k 2k 1, which is impossible. Thus, p = k+2. Hence p = k+2 for some positive integer k, and so 2(k) 2(k+2) 2. Hence, 2k γ cc (T v 1 v 2 ) 2k and so T v 1 v 2 τ. Now we are ready to prove the following theorem. Theorem 5.10 For any tree T = K 1,p 1 with p 6, τ = 2 i=0 X i. Proof. Suppose T X 0. If T = P 6, then γ cc (T) = 4 = 2p(T). If T is not P 6 then by Proposition 5.1, γ cc (T) = 2p(T) and so T τ. Suppose T X 1. Then p = p(t) = k + 1 for some positive integer k, and T can be constructed from T X 0 by applying operation 5.2 to T. Let u be the P 1 attached. By Proposition 5.1, D(T ) is a complete cototal dominating set of cardinality 2k. But then D {u} is a complete cototal dominating set of T of cardinality 2k + 1. Therefore, γ cc (T) 2(k+1) = 2k +1 and so γ cc (T) = 2p. Hence T τ. 77
Suppose T X 2, then p = p(t) = k + 2 for some positive integer k, and T can be constructed from T X 0 by applying operation 5.. Let u and u be the added vertices. By Proposition 5.1, D(T ) is a complete cototal dominating set of cardinality 2k. But then D {u,u } is a complete cototal dominating set of T of cardinality 2k +2. Therefore γ cc (T) 2(k+2) = 2k +2, and so γ cc (T) = 2p. Hence T τ. We now show that τ 2 i=0 X i. To prove this we apply induction on the number of vertices p of the tree T τ. If p = 6, then T = P 6 X 0. Suppose T τ has order p = p(t) 1 and assume that T is a tree with 6 p(t ) < p and T τ. Then T 2 i=0 X i. Let D be any complete cototal dominating set. Then each vertex in V D is adjacent to at least one vertex in D. Furthermore, every vertex in D is either a degree two support vertex adjacent to exactly one vertex in V D or a pendant vertex adjacent to degree two support vertex that is adjacent to exactly one vertex in V D. Hence the number of vertices in a tree is the multiples of three and p = k for some positive integer k 2. Let v be the end point of the diametrical path v = v 1,v 2,,v i in V D. As T contains no cycles, v is a pendant vertex of V D. Thus deg(v) = 2 78
and V D 2, and so v 2 is adjacent to a degree two support vertex. If V D = 2, then T = P 6 X 0. We therefore assume that V D. Note that v 2 is neither a pendant vertex nor a support vertex of T = T v 1 u 0 u 1. Moreover, D {u 0,u 1 } is a complete cototal dominating set of T and so 2(k ) γ cc (T ) 2(k) 2, which implies that 2k 2 γ cc (T ) 2k 2 and so T τ. By our induction assumption T X 0. Hence T X 0. Next, we have the following characterization of graphs for which the complete cototal domination number is equal to the total domination number. Theorem 5.11 For any graph G, γ cc (G) = γ t (G) if and only if G has a total dominating set D such that V D has no isolated vertices. In the next theorem we prove the relation between connected domination number and complete cototal domination number. Theorem 5.12 If Δ(G) = p 1, then γ cc (G) 1+γ c (G). Proof. If Δ(G) = p 1. Then clearly, γ c (G) = 1. By the definition of complete cototaldominationand by Theorem 5., γ cc (G) 2. Therefore, clearly γ cc (G) γ c (G)+1. 79
Next theorem gives the relation between cototal domination number and the complete cototal domination number. Theorem 5.1 For any graph G, a cototal dominating set D is said to be a complete cototal dominating set of G if D has no isolated vertices. Next, we obtain Nordhaus-Gaddum type results for complete cototal domination number of G. Theorem 5.14 Let G be any nontrivial connected graph. If both G and G has no isolated vertices, then (i) γ cc (G)+γ cc (G) 2p (ii) γ cc (G) γ cc (G) p 2. Further, equality holds for G = P 4 or C 5. Finally, we conclude this chapter by exploring the following two open problems for the further research on this topic. Open Problem 5.1. Characterize the graphs for which γ cc (G) = γ cl (G). Open Problem 5.2. Characterize the graphs for which γ cc (G) = 1+γ c (G). 80