ON THE INJECTIVE DOMINATION OF GRAPHS

Palestine Journal of Mathematics Vol. 7(1)(018), 0 10 Palestine Polytechnic Uniersity-PPU 018 ON THE INJECTIVE DOMINATION OF GRAPHS Anwar Alwardi, R. Rangarajan and Akram Alqesmah Communicated by Ayman Badawi MSC 010 Classications: 0C9. Keywords and hrases: Injectie domination number, Injectie indeendence number, Injectie domatic number. Abstract. Let G = (V, E) be a simle grah. A subset D of V is called injectie dominating set (Inj-dominating set) if for eery ertex V D there exists a ertex u D such that G(u, ) 1, where G(u, ) is the number of common neighbors between the ertices u and. The minimum cardinality of such dominating set denoted by γ in (G) and is called injectie domination number (Inj-domination number) of G. In this aer, we introduce the injectie domination of a grah G and analogous to that, we dene the injectie indeendence number (Inj-indeendence number) β in (G) and injectie domatic number (Inj-domatic number) d in (G). Bounds and some interesting results are established. 1 Introduction By a grah we mean a nite, undirected with no loos and multile edges. In general, we use X to denote the subgrah induced by the set of ertices X and N(), N[] denote the oen and closed neighborhood of a ertex, resectiely. The distance between two ertices u and in G is the number of edges in a shortest ath connecting them, this is also known as the geodesic distance. The eccentricity of a ertex is the greatest geodesic distance between and any other ertex and denoted by e(). A set D of ertices in a grah G is a dominating set if eery ertex in V D is adjacent to some ertex in D. The domination number γ(g) is the minimum cardinality of a dominating set of G. We denote to the smallest integer greater than or equal to x by x and the greatest integer less than or equal to x by x. A strongly regular grah with arameters (n, k, λ, µ) is a grah with n ertices such that the number of common neighbors of two ertices u and is k, λ or µ according to whether u and are equal, adjacent or non-adjacent, resectiely. When λ = 0 the strongly regular grah is called strongly regular grah with no triangles (SRNT grah). A strongly regular grah G is called rimitie if G and G are connected. For terminology and notations not secically dened here we refer the reader to []. For more details about domination number and neighborhood number and their related arameters, we refer to [3], [4]. The common neighborhood domination in grah has introduced in []. A subset D of V is called common neighborhood dominating set (CN-dominating set) if for eery ertex V D there exists a ertex u D such that u E(G) and G(u, ) 1, where G(u, ) is the number of common neighborhood between the ertices u and. The minimum cardinality of such dominating set denoted by γ cn (G) and is called common neighborhood domination number (CN-domination number) of G. The common neighborhood (CN-neighborhood) of a ertex u V (G) denoted by N cn (u) is dened as N cn (u) = { V (G) : u E(G) and G(u, ) 1}. The common neighborhood grah (congrah) of G, denoted by con(g), is the grah with the ertex set 1,,...,, in which two ertices are adjacent if and only if they hae at least one common neighbor in the grah G [1]. In this aer, we introduce the concet of injectie domination in grahs. In ordinary domination between ertices is enough for a ertex to dominate another in ractice. If the ersons

ON THE INJECTIVE DOMINATION OF GRAPHS 03 hae common friend then it may result in friendshi. Human beings hae a tendency to moe with others when they hae common friends. Injectie Dominating Sets In defence and domination roblem in some situations there should not be direct contact between two indiiduals but can be linked by a third erson this motiated us to introduced the concet of injectie domination. Denition.1 ([1]). Let G be simle grah with ertex set V (G) = { 1,,..., }. For i j, the common neighborhood of the ertices i and j, denoted by G( i, j ), is the set of ertices, different from i and j, which are adjacent to both i and j. Denition.. Let G = (V, E) be a grah. A subset D of V is called injectie dominating set (Inj-dominating set) if for eery ertex V either D or there exists a ertex u D such that G(u, ) 1. The minimum cardinality of Inj-dominating set of G denoted by γ in (G) and called injectie domination number (Inj-domination number) of G. For examle consider a grah G in Figure 1. Then {, 7} is a minimum dominating set, {, 3, 4,, } is a minimum CN-dominating set and {1} is a minimum injectie dominating set of G. Thus γ(g) =, γ cn (G) = and γ in (G) = 1. 1 3 7 4 Figure 1. Grah with γ in (G) = 1 Obiously, for any grah G, the ertex set V (G) is an Inj-dominating set, that means any grah G has an Inj-dominating set and hence Inj-domination number. An injectie dominating set D is said to be a minimal Inj-dominating set if no roer subset of D is an Inj-dominating set. Clearly each minimum Inj-dominating set is minimal but the conerse is not true in general, for examle let G be a grah as in Figure 1. Then {, 3} is a minimal Inj-dominating set but not minimum Inj-dominating set. Let u V. The Inj-neighborhood of u denoted by N in (u) is dened as N in (u) = { V (G) : G(u, ) 1}. The cardinality of N in (u) is called the injectie degree of the ertex u and denoted by deg in (u) in G, and N in [u] = N in (u) {u}. The maximum and minimum injectie degree of a ertex in G are denoted resectiely by Din(G) and δ in (G). That is Din(G) = max u V N in (u), δ in (G) = min u V N in (u). The injectie comlement of G denoted by G inj is the grah with same ertex set V (G) and any two ertices u and in G inj are adjacent if and only if they are not Inj-adjacent in G. A subset S of V is called an injectie indeendent set (Inj-indeendent set), if for eery u S, / N in (u) for all S {u}. An injectie indeendent set S is called maximal if any ertex set roerly containing S is not Injindeendent set, the maximum cardinality of Inj-indeendent set is denoted by β in, and the lower Inj-indeendence number i in is the minimum cardinality of the Inj-maximal indeendent set. As usual P, C, K and W are the -ertex ath, cycle, comlete, and wheel grah resectiely, K r,m is the comlete biartite grah on r + m ertices and S is the star with ertices.

04 Anwar Alwardi, R. Rangarajan and Akram Alqesmah Proosition.3. Let G = (V, E) be a grah and u V be such that G(u, ) = 0 for all V (G). Then u is in eery injectie dominating set, such ertices are called injectie isolated ertices. Proosition.4. Let G = (V, E) be strongly regular grah with arameters (n, k, λ, µ). Then γ in (G) = 1 or. Proosition.. For any grah G, γ in (G) γ cn (G) Proof. From the denition of the CN-dominating set of a grah G. For any grah G any CNdominating set D is also Inj-dominating set. Hence γ in (G) γ cn (G). We note that the Inj-domination number of a grah G may be greater than, smaller than or equal to the domination number of G. Examle.. (i) γ in (P ) = ; γ(p ) = 1. (ii) γ in (C ) = ; γ(c ) =. (iii) If G is the Petersen grah, then γ in (G) = ; γ(g) = 3. Proosition.7. (i) For any comlete grah K, where, γ in (K ) = 1. (ii) For any wheel grah G = W, γ in (G) = 1. (iii) For any comlete biartite grah K r,m, γ in (K r,m ) =. (i) For any grah G, γ in (K + G) = 1, where. Proosition.8. For any grah G with ertices, 1 γ in (G). Proosition.9. Let G be grah with ertices. Then γ in (G) = if and only if G is a forest with D(G) 1. Proof. Let G be a forest with D(G) 1. Then we hae two cases. Case 1. If G is connected, then either G = K or G = K 1. Hence, γ in (G) =. Case. If G is disconnected, then G = n 1 K n K 1, thus γ in (G) =. Conersely, if γ in (G) =, then all the ertices of G are Inj-isolated, that means G is isomorhic to K 1 or K or to the disjoint union of K 1 and K, that is G = n 1 K n K 1 for some n 1,n {0, 1,,...}. Hence, G is a forest with D(G) 1. Proosition.10. Let G be a nontriial connected grah. Then γ in (G) = 1 if and only if there exists a ertex V (G) such that N() = N cn () and e(). Proof. Let V (G) be any ertex in G such that N() = N cn () and e(). Then for any ertex u V (G) {} if u is adjacent to, since N() = N cn (), then obious u N in (). If u is not adjacent to, then G(u, ) 1. Thus for any ertex u V (G) {}, G(u, ) 1. Hence, γ in (G) = 1. Conersely, if G is a grah with ertices and γ in (G) = 1, then there exist at least one ertex V (G) such that deg in () = 1, then any ertex u V (G) {} either contained in a triangle with or has distance two from. Hence, N() = N cn () and e(). Theorem.11 ([]). For any ath P and any cycle C, where 3, we hae, γ(p ) = γ(c ) =. 3 Proosition.1 ([]). For any ath P and any cycle C (i) con(p ) = P P.

(ii) con(c ) = ON THE INJECTIVE DOMINATION OF GRAPHS 0 C, if is odd and 3; P P, if = 4; C C, if is een. From the denition of the common neighborhood grah and the Inj-domination in a grah the following roosition can easily eried. Proosition.13. For any grah G, γ in (G) = γ(con(g)). The roof of the following roosition is straightforward from Theorem.11 and Proosition.1. Proosition.14. For any cycle C with odd number of ertices 3, γ in (C ) = γ(c ) = 3 Theorem.1. For any cycle C with een number of ertices 3, γ in (C ) =. Proof. From Proosition.13, Theorem.11 and Proosition.1, if is een, then γ in (C ) = γ((c ) (C )) = γ(c ) =. Proosition.1. For any odd number > 3, + 1 γ in (P ) = + 1 Proof. From Proosition.13, Theorem.11 and Proosition.1, if is odd then, + 1 1 γ in (P ) = γ(p P ) = γ(p +1 P 1 ) = +.. Proosition.17. For any een number 4, γ in (P ) =. Proof. From Proosition.13, Theorem.11 and Proosition.1, if is een then, = =. Hence γ in(p ) =. Theorem.18. Let G = (V, E) be a grah without Inj-isolated ertices. Inj-dominating set, then V D is an Inj-dominating set. If D is a minimal Proof. Let D be a minimal Inj-dominating set of G. Suose V D is not Inj-dominating set. Then there exists a ertex u in D such that u is not Inj-dominated by any ertex in V D, that is G(u, ) = 0 for any ertex in V D. Since G has no Inj-isolated ertices, then there is at least one ertex in D {u} has common neighborhood with u. Thus D {u} is Inj-dominating set of G, which contradicts the minimality of the Inj dominating set D. Thus eery ertex in D has common neighborhood with at least one ertex in V D. Hence V D is an Inj-dominating set. Theorem.19. Let G be a grah. Then the injectie dominating set D is minimal if and only if for eery ertex D, one of the following conditions holds (i) is Inj-isolated ertex. (ii) There exists a ertex u in V D such that N in (u) D = {}.

0 Anwar Alwardi, R. Rangarajan and Akram Alqesmah Proof. Suose D is a minimal Inj-dominating set of G. Then D {} is not Inj-dominating set, then there exists at least one ertex u (V D) {} is not Inj-dominated by any ertex in D, so we hae two cases. Case 1. If u D, then u is Inj-isolated ertex. Case. If u V D, then u has common neighborhood with only one ertex in D, that means N in (u) D = {}. Conersely, suose D is an Inj-dominating set of G and for each ertex D one of the two conditions holds, we want to roe that D is a minimal Inj-dominating set. Suose that D is not minimal. Then there is at least one ertex D such that D {} is an Inj-dominating set. Thus has common neighborhood with at least one ertex in D {}. Hence, condition (i) is not hold. Also, V D is an Inj-dominating set, then eery ertex in V D has common neighborhood with at least one ertex in D {}. Therefore condition (ii) is not hold. Hence, neither condition (i) nor condition (ii) holds, which is a contradiction. Theorem.0. A grah G has a unique minimal Inj-dominating set if and only if the set of all Inj-isolated ertices forms an Inj-dominating set. Proof. Let G has a unique minimal Inj-dominating set D, and suose S = {u V : u is Inj-isolated ertex}. Then S D. Now suose D S, let D S, since is not Injisolated ertex, then V {} is an Inj-dominating set. Hence there is a minimal Inj-dominating set D 1 V {} and D 1 D a contradiction to the fact that G has a unique minimal Injdominating set. Therefore S = D. Conersely, if the set of all Inj-isolated ertices in G forms an Inj-dominating set, then it is clear that G has a unique minimal Inj-dominating set. Theorem.1. For any (, q)-grah G, γ in (G) q. Proof. Let D be a minimum Inj-dominating set of G. Since eery ertex in V D has common neighborhood with at least one ertex of D, then q V D. Hence, γ in (G) q. Theorem.. For any grah G with ertices, 1+D in (G) γ in(g). Further, the equality holds if and only if for eery minimum Inj-dominating set D in G the following conditions are satised: (i) for any ertex in D, deg in () = Din(G); (ii) D is Inj-indeendent set in G; (iii) eery ertex in V D has common neighborhood with exactly one ertex in D. Proof. Let S be any minimum Inj-dominating set in G. Clearly each ertex in G will Injdominate at most (Din(G) + 1) ertices, so = N in [S] γ in (G)(Din(G) + 1), hence 1+D in (G) γ in(g). Therefore 1+D in (G) γ in(g). Suose the gien conditions are hold for any minimum Inj-dominating set D in G. Then obiously γ in (G)Din(G) + γ in (G) =. Hence, 1+D = γ in(g) in(g). Conersely, suose the equality holds, and suose that one from the conditions is not satised. Then < γ in (G)Din(G) + γ in (G), a contradiction. Examle.3. Let G = C, where 3 and is odd number. Then the equality in Theorem. is hold. Since 1+D = and by Proosition.14 we hae γ in(g) 3 in(g) =. 3 Theorem.4. Let G be a grah on ertices and δ in (G) 1. Then γ in (G). Proof. Let D be any minimal Inj-dominating set in G. Then by Theorem.18, V D is also an Inj-dominating set in G. Hence, γ in (G) min{ D, V D } /. Theorem.. For any grah G on ertices, γ in (G) Din(G) Proof. Let be a ertex in G such that deg in () = Din(G). Then has common neighborhood with N in () = Din(G) ertices. Thus, V N in () is an Inj-dominating set. Therefore γ in (G) V N in (). Hence, γ in (G) Din(G).

ON THE INJECTIVE DOMINATION OF GRAPHS 07 Proosition.. For any grah G with diameter less than or equal three and maximum degree D(G), γ in (G) D(G) + 1. Proof. Let diam(g) 3 and V (G) such that deg() = D(G). Clearly that, if diam(g) = 1, then G is a comlete grah and the result holds. Suose diam(g) = or 3. Let V i (G) V (G) be the sets of ertices of G which hae distance i from, where i = 1,, 3. Obiously, the set S = V 1 (G) {} is an Inj-dominating set of G of order D(G)+1. Hence, γ in (G) D(G)+1. The Cartesian roduct G H of two grahs G and H is a grah with ertex set V (G) V (H) and edge set E(G H) = {((u, u ), (, )) : u = and (u, ) E(H), or u = and (u, ) E(G)}. 1 3 4 m u 1 u u 3 u 4 u m Figure. P m P Proosition.7. For any grah G = P m P, γ in (G) = m. Proof. Let G = P m P. From Figure, it is easy to see that any two adjacent ertices i, u i can Inj-dominate all the ertices of distance less than or equal two from i or u i, then γ in (G) m. obiously from Figure, D in(g) = 4, then by Theorem., γ in (G) m. Now, if m < or m 0 (mod ), then m = m and hence γ in(g) = m. Otherwise, m = m + 1, but in this case the equality of Theorem. does not hold because the third condition is not satised. Hence, γ in (G) = m. Proosition.8. For any grah G = C m P, γ in (G) = m. Proof. The roof is same as in Proosition.7. The Comosition G H or G[H] has its ertex set V (G) V (H), with (u, u ) is adjacent to (, ) if either u is adjacent to in G or u = and u is adjacent to in H. Proosition.9. For any grah G isomorhic to P m P n or P m C n or C m P n or C m C n, γ in (G) = m. Proof. Let G be a grah isomorhic to P m P n or P m C n or C m P n or C m C n. Then from the denition of the Comosition roduct, N(w) = N cn (w), w V (G), then each ertex w = (u, ) in G Inj-dominates its neighbors and all the ertices of distance two of it, then γ in (G) m, but in this grah D in(g) = n 1, so by Theorem., γ in (G) mn n = m. Hence, γ in (G) = m. Denition.30. Let G = (V, E) be a grah. S V (G) is called Inj-indeendent set if no two ertices in S hae common neighbor. An Inj-indeendent set S is called maximal Injindeendent set if no suerset of S is Inj-indeendent set. The Inj-indeendent set with maximum size called the maximum Inj-indeendent set in G and its size called the Inj-indeendence number of G and denoted by β in (G). Theorem.31. Let S be a maximal Inj-indeendent set. Then S is a minimal Inj-dominating set.

08 Anwar Alwardi, R. Rangarajan and Akram Alqesmah Proof. Let S be a maximal Inj-indeendent set and u V S. If u / N in () for eery S, then S {u} is an Inj-indeendent set, a contradiction to the maximality of S. Therefore u N in () for some S. Hence, S is an Inj-dominating set. To roe that S is a minimal Inj-dominating set, suose S is not minimal. Then for some u S the set S {u} is an Inj-dominating set. Then there exist some ertex in S has a common neighborhood with u, a contradiction because S is an Inj-indeendent set. Therefore S is a minimal Inj-dominating set. Corollary.3. For any grah G, γ in (G) β in (G). 3 Injectie domatic number in a grah Let G = (V, E) be a grah. A artition D of its ertex set V (G) is called a domatic artition of G if each class of D is a dominating set in G. The maximum order of a artition of V (G) into dominating sets is called the domatic number of G and is denoted by d(g). Analogously as to γ(g) the domatic number d(g) was introduced, we introduce the injectie domatic number d in (G), and we obtain some bounds and establish some roerties of the injectie domatic number of a grah G. Denition 3.1. Let G = (V, E) be a grah. A artition D of its ertex set V (G) is called an injectie domatic (in short Inj-domatic) artition of G if each class of D is an Inj-dominating set in G. The maximum order of a artition of V (G) into Inj-dominating sets is called the Injdomatic number of G and is denoted by d in (G). For eery grah G there exists at least one Inj-domatic artition of V (G), namely {V (G)}. Therefore d in (G) is well-dened for any grah G. Theorem 3.. (i) For any comlete grah K, d in (K ) = d cn (K ) = d(k ) =. (ii) d in (G) = 1 if and only if G has at least one Inj-isolated ertex. (iii) For any wheel grah of ertices, d in (W ) =. (i) For any comlete biartite grah K r,m, d in (K r,m ) = { min{r, m}, if r, m ; 1, otherwise. () For any grah G, if N in () = N() for any ertex in V (G), then d in (G) = d(g). Proof. (i) If G = (V, E) is the comlete grah K, then for any ertex the set {} is a minimum CN-dominating set and also a minimum Inj-dominating set. Then the maximum order of a artition of V (G) into Inj-dominating or CN-dominating sets is. Hence, d in (K ) = d cn (K ) =. (ii) Let G be a grah which has an Inj-isolated ertex say, then eery Inj-dominating set of G must contain the ertex. Then d in (G) = 1. Conersely, if d in (G) = 1 and suose G has no Inj-isolated ertex, then by Theorem.4, γ in (G), so if we suose D is a minimal Inj-dominating set in G, then V D is also a minimal Inj-dominating set. Thus d in (G), a contradiction. Therefore G has at least one Inj-isolated ertex. (iii) Since for eery ertex of the wheel grah the deg in () = 1. Hence, d in (W ) =. (i) and () the roof is obious.

ON THE INJECTIVE DOMINATION OF GRAPHS 09 Eidently each CN-dominating set in G is an Inj-dominating set in G, and any CN-domatic artition is an Inj-domatic artition. We hae the following roosition. Proosition 3.3. For any grah G, d in (G) d cn (G). Theorem 3.4. For any grah G with ertices, d in (G) γ in (G). Proof. Assume that d in (G) = d and {D 1, D,..., D d } is a artition of V (G) into d numbers of Inj-dominating sets, clearly D i γ in (G) for i = 1,,..., d and we hae = d D i=1 i dγ in (G). Hence, d in (G) γ in(g). Theorem 3.. For any grah G with ertices, d in (G). δ in(g) Proof. Let D be any subset of V (G) such that D δ in (G). For any ertex V D, we hae N in [] 1 + δ in (G). Therefore N in () D ϕ. Thus D is an Inj-dominating set of G. So, we can take any δ in (G) disjoint subsets each of cardinality δin (G). Hence, d in (G) δ in (G). Theorem 3.. For any grah G with ertices d in (G) δ in (G) + 1. Further the equality holds if G is comlete grah K. Proof. Let G be a grah such that d in (G) > δ in (G) + 1. Then there exists at least δ in (G) + Inj-dominating sets which they are mutually disjoint. Let be any ertex in V (G) such that deg in () = δ in (G). Then there is at least one of the Inj-dominating sets which has no intersection with N in []. Hence, that Inj-dominating set can not dominate, a contradiction. Therefore d in (G) δ in (G) + 1. It is obious if G is comlete, then d in (G) = δ in (G) + 1. Theorem 3.7. For any grah G with ertices, d in (G) + d in (G inj ) + 1 Proof. From Theorem 3., we hae d in (G) δ in (G) + 1 and d in (G inj ) δ in (G inj ) + 1, and clearly δ in (G inj ) = 1 Din(G). Hence, d in (G) + d in (G inj ) δ in (G) + Din(G) + 1 + 1. Theorem 3.8. For any grah G with ertices and without Inj-isolated ertices, d in (G) + γ in (G) + 1. Proof. Let G be a grah with ertices. Then by Theorem., we hae γ in (G) Din(G) δ in (G), and also from Theorem 3., d in (G) δ in (G) + 1. Then d in (G) + γ in (G) δ in (G) + 1 + δ in (G). Hence, d in (G) + γ in (G) + 1.

10 Anwar Alwardi, R. Rangarajan and Akram Alqesmah References [1] A. Alwardi, B. Arsit'c, I. Gutman and N. D. Soner, The common neighborhood grah and its energy, Iran. J. Math. Sci. Inf., 7(), 1 8 (01). [] Anwar Alwardi, N. D. Soner and Karam Ebadi, On the common neighbourhood domination number, Journal of Comuter and Mathematical Sciences, (3), 47 (011). [3] S. Arumugam, C. Siagnanam, Neighborhood connected and neighborhood total domination in grahs, Proc. Int. Conf. on Disc. Math., 334 B. Chaluaraju, V. Lokesha and C. Nandeesh Kumar Mysore 4 1 (008). [4] B. Chaluaraju, Some arameters on neighborhood number of a grah, Electronic Notes of Discrete Mathematics, Elseier, 33 139 14 (009). [] F. Harary, Grah theory, Addison-Wesley, Reading Mass. (199). [] T. W. Haynes, S. T. Hedetniemi and P. J. Slater, Fundamentals of domination in grahs, Marcel Dekker, Inc., New York (1998). [7] S. M. Hedetneimi, S. T. Hedetneimi, R. C. Laskar, L. Markus and P. J. Slater, Disjoint dominating sets in grahs, Proc. Int. Conf. on Disc.Math., IMI-IISc, Bangalore 88 101 (00). [8] V. R. Kulli and S. C. Sigarkanti, Further results on the neighborhood number of a grah, Indian J. Pure and Al. Math. 3(8) 7 77 (199). [9] E. Samathkumar and P. S. Neeralagi, The neighborhood number of a grah, Indian J. Pure and Al. Math. 1() 1 13 (198). [10] H. B. Walikar, B. D. Acharya and E. Samathkumar, Recent deelomentsin the theory of domination in grahs, Mehta Research institute, Allahabad, MRI Lecture Notes in Math. 1 (1979). Author information Anwar Alwardi, Deartment of Mathematics, College of Education, Yafea, Uniersity of Aden, Yemen. E-mail: a wardi@hotmail.com R. Rangarajan, Deartment of Studies in Mathematics, Uniersity of Mysore, Mysore 70 00, India. E-mail: rajra3@gmail.com Akram Alqesmah, Deartment of Studies in Mathematics, Uniersity of Mysore, Mysore 70 00, India. E-mail: aalqesmah@gmail.com Receied: June 17, 01. Acceted: March 1, 017.