ON THE INVERSE SIGNED TOTAL DOMINATION NUMBER IN GRAPHS. D.A. Mojdeh and B. Samadi

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1 Opuscula Math. 37, no. 3 (017), Opuscula Mathematica ON THE INVERSE SIGNED TOTAL DOMINATION NUMBER IN GRAPHS D.A. Mojdeh and B. Samadi Communicated by Dalibo Fonček Abstact. In this pape, we study the invese signed total domination numbe in gaphs and pesent new shap lowe and uppe bounds on this paamete. Fo example by making use of the classic theoem of Tuán (1941), we pesent a shap uppe bound on K +1-fee gaphs fo. Also, we bound this paamete fo a tee fom below in tems of its ode and the numbe of leaves and chaacteize all tees attaining this bound. Keywods: invese signed total dominating function, invese signed total domination numbe, k-tuple total domination numbe. Mathematics Subject Classification: 05C INTRODUCTION Thoughout this pape, let G be a finite connected gaph with vetex set V = V (G), edge set E = E(G), minimum degee δ = δ(g) and maximum degee = (G). Fo any vetex v V, N(v) = u G uv E} denotes the open neighbohood of v in G, and N[v] = N(v) v} denotes its closed neighbohood. Fo all A, B V let [A, B] be the set of edges having one end point in A and the othe in B. We use [11] as a efeence fo teminology and notation which ae not defined hee. A set D V is a total dominating set in G if each vetex in V is adjacent to at least one vetex in D. The total domination numbe γ t (G) is the minimum cadinality of a total dominating set in G. The k-tuple total dominating set (o k-total dominating set) was intoduced by Kulli [5], as a subset D V with N(v) D k fo all v V, whee 1 k δ. The k-tuple total domination numbe (o k-total domination numbe), denoted γ k,t (G), is the smallest numbe of vetices in a k-tuple total dominating set. (We should note that, k-tuple total domination numbe is diffeent fom total k-distance domination numbe [6]). Clealy, γ 1,t (G) = γ t (G). c Wydawnictwa AGH, Kakow

2 448 D.A. Mojdeh and B. Samadi Let B V. Fo a eal-valued function f : V R we define f(b) = v B f(v). Also, f(v ) is the weight of f. A signed total dominating function, abbeviated STDF, of G is defined in [1] as a function f : V 1, 1} such that f(n(v)) 1 fo evey v V. The signed total domination numbe (STDN) of G, γ st (G), is the minimum weight of a STDF of G. If we eplace and minimum with and maximum, espectively, in the definition of STDN, we will have the signed total -independence function (STIF) and the signed total -independence numbe (STIN) of the gaph, denoted by α st(g). This concept was intoduced in [10] and studied in [8, 9] as the negative decision numbe. An invese signed total dominating function, abbeviated ISTDF, of G is defined in [4] as a function f : V 1, 1} such that f(n(v)) 0 fo evey v V. The invese signed total domination numbe (ISTDN), denoted by γ 0 st(g), is the maximum weight of an ISTDF of G. Fo moe infomation the eade can consult [1]. In this pape, we continue the study of the concept of invese signed total domination in gaphs. In Section, we pesent a shap uppe bound on a geneal gaph G by consideing the concept of k-tuple total domination in gaphs. Moeove, as an application of the well-known theoem of Tuán [7] about K p -fee gaphs we give an uppe bound on γ 0 st(g) fo a K +1 -fee gaph G and show the bound is the best possible by constucting an -patite gaph attaining the bound. In Section 3, we discuss ISTDN fo egula gaphs and give shap lowe and uppe bounds on the ISTDN of egula gaphs. Finally, in Section 4 we show that this paamete is not bounded fom both above and below in geneal, even fo tees. Also, we give a lowe bound on the ISTDN of a tee T as ( γst(t 0 l1 ls ) ) n , whee l 1,..., l s ae the numbe of leaves adjacent to its s suppot vetices, and chaacteize all tees attaining this bound.. TWO UPPER BOUNDS Thoughout this pape, if f is an ISTDF of G, then we let P and M denote the sets of those vetices which ae assigned 1 and 1 unde f, espectively. We apply the concept of tuple total domination to obtain a shap uppe bound on γ st (G). It is easy to check the poof of the following obsevation. Obsevation.1. Let K n and C n be complete gaph and cycle with n vetices. Then (i) γst(k 0 if n 0 (mod ), n ) = 1 if n 1 (mod ); ([4]) 0 if n 0 (mod 4) (ii) γst(c 0 n ) = 1 if n 1 o 3 (mod 4), if n (mod 4); ([4])

3 On the invese signed total domination numbe in gaphs 449 (iii) γ t (K n ) = fo n ; n (iv) γ t (C n ) = + 1 if n (mod 4), n othewise. We make use of them to show that the following bound is shap. Theoem.. If G is a gaph of ode n and minimum degee δ 1, then γst(g) 0 γt (G) + δ n and this bound is shap. Poof. Let f : V 1, 1} be a maximum ISTDF of G. The condition f(n(v)) 0, fo all v V, leads to N(v) M deg(v) δ. So, M is a δ -tuple total dominating set in G and theefoe (n γst(g))/ 0 = M γ δ,t(g). (.1) Now let D be a minimum δ -tuple total dominating set in G. Hence, N(v) D δ, fo all v V. Let u D. Then N(v) (D \ u}) δ 1, fo all v V. This shows that D \ u} is a ( δ 1)-tuple total dominating set in G. Theefoe, γ δ,t(g) γ ( δ 1),t(G) + 1. If we iteate this pocess, we finally aive at δ δ γ δ,t(g) γ ( δ 1),t(G) γ 1,t(G) + 1 = γ t (G) + 1. Thus, (.1) yields δ (n γst(g))/ 0 γ t (G) + 1 as desied. Moeove, this bound is shap. It is sufficient to conside the complete gaph K n, when n and the cycle C n, when n 3. Note that the diffeence between γst(g) 0 and n γt(g)+δ may be lage. It is easy to check that, fo a complete bipatite gaph K m,n, 0 if n, m 0 (mod ), γst(k 0 m,n ) = 1 if m and n have diffeent paity, if n, m 1 (mod ). While n+m γt(km,n)+δ n 3, whee n = maxm, n}, the uppe bound in Theoem.4 woks bette fo bipatite gaphs. We fist ecall that a gaph is K p -fee if it does not contain the complete gaph K p as an induced subgaph. Fo ou next uppe bound, we use the following well-known theoem of Tuán [7]. Theoem.3. If G is a K +1 -fee gaph of ode n, then E(G) 1 n.

4 450 D.A. Mojdeh and B. Samadi Theoem.4. Let be an intege, and G be a K +1 -fee gaph of ode n. If c = δ, then ( γst(g) 0 n ) c + c cn and this bound is shap. Poof. Let f be an ISTDF of G. Let v P. Since f(n(v)) 0, N(v) M δ. Theefoe δ [M, P ] P. (.) Futhemoe, Theoem.3 implies [M, P ] = v M N(v) P v M Combining this inequality chain and (.), we aive at N(v) M = E(G[M]) 1 M. 1 M + c M cn 0. Solving the above inequality fo M we obtain ( ) M c + c ( 1) cn. Because of M = (n γst(g))/, 0 we obtain the desied uppe bound. That the bound is shap, can be seen by constucting an -patite gaph attaining this uppe bound as follows. Let H i be a complete bipatite gaph with vetex patite sets X i and Y i, whee X i = 1 and Y i = ( 1) fo all 1 i. Let the gaph H() be the disjoint union of H 1,..., H by joining each vetex of X i (in H i ) with all vetices in union of X j, i j. Also, we add ( 1) 3 edges between Y i and the union of Y j, i j, so that evey vetex of Y i has exactly 1 neighbos in this union. Now let Z i = X i Y i+1, fo all 1 i (mod ). Then H() is an -patite gaph of ode n = ( 1) with patite sets Z 1,..., Z. Clealy, evey vetex in Y i has the minimum degee δ = and hence c = 1. Now we define f : V (H()) 1, 1}, by 1 if v X1... X f(v) =, 1 if v Y 1... Y. It is easy to check that f is an ISTDF of H() with weight ( ( 1) ( 1) = n ) c + c cn. This completes the poof.

5 On the invese signed total domination numbe in gaphs REGULAR GRAPHS Ou aim in this section is to give shap lowe and uppe bounds on the ISTDN of a egula gaph. Henning [] and Wang [9] poved that fo an -egula gaph G, ( ++ γ st (G) +3 )n if 0 (mod ), ( )n if 1 (mod ) (3.1) (see []), and α st(g) ( 1 1+ )n if 0 (mod ), ( )n if 1 (mod ) (3.) (see [9]). Futhemoe, these bounds ae shap. Also, the following shap lowe and uppe bounds on STDN and STIN of an -egula gaph G can be found in [1] and [8, 10], espectively. γ st (G) n/ if 0 (mod ), n/ if 1 (mod ) (3.3) and α st(g) 0 if 0 (mod ), n/ if 1 (mod ). (3.4) Applying the concept of tuple total domination we can show that thee ae special elationships among γ 0 st(g), γ st (G) and α st(g) when we estict ou discussion to the egula gaph G. Theoem 3.1. Let G be an -egula gaph. If is odd, then γ 0 st(g) = γ st (G) and if is even, then γ 0 st(g) = α st(g). Poof. By (.1), we have γ 0 st(g) n γ,t (G). (3.5) Let D be a minimum -tuple total dominating set in G. We define f : V 1, 1}, by 1 if v D, f(v) = 1 if v V \ D. Taking into account that D is a -tuple total dominating set, we have f(n(v)) = N(v) (V \ D) N(v) D = deg(v) N(v) D 0. Thus, f is an ISTDF of G with weight n γ,t (G) and theefoe γ 0 st(g) n γ,t (G). Now the inequality (3.5) implies γ 0 st(g) = n γ,t (G). (3.6)

6 45 D.A. Mojdeh and B. Samadi The following equalities fo the STDN and the STIN of G can be poved similaly: and Fom (3.6), (3.7) and (3.8), the desied esults follow. γ st (G) = γ +1,t(G) n (3.7) α st(g) = n γ,t (G). (3.8) Using Theoem 3.1 and inequalities (3.1) (3.4) we can bound γ 0 st(g) of a egula gaph G fom both above and below as follows. Theoem 3.. Let G be an -egula gaph of ode n. Then and ( ( 1 ) n γ st(g) 0, 0 (mod ), Futhemoe, these bounds ae shap. ) n γ 0 st(g) n/, 1 (mod ). As an immediate esult of Theoem 3.1 we have γ 0 st(g) = γ st (G) fo all cubic gaph G. Hosseini Moghaddam et al. [3] showed that γ st (G) n/3 is a shap uppe bound fo all connected cubic gaph G diffeent fom Heawood gaph G 14 (see Figue 1). Theefoe, if G is a connected cubic gaph diffeent fom G 14, then γ 0 st(g) n/3 is a shap lowe bound. Fig. 1. The Heawood gaph G TREES It has been poved by Atapou et al. in [1] that γ 0 st(t ) (n 4)/3, fo any tee T of ode n. On the othe hand, Henning [] poved that γ st (T ), fo any tee T of ode at least two. Theefoe, γ st (T ) is bounded fom below by the lowe bound, not depending on the ode o anything else, fo any tee T. A simila esult cannot be pesented fo γ 0 st(t ). In what follows we show that in geneal γ 0 st(t ) is not bounded fom both above and below. In fact, we pove a stonge esult as follows.

7 On the invese signed total domination numbe in gaphs 453 Poposition 4.1. Fo any intege k, thee exists a tee T with γ 0 st(t ) = k. Poof. We conside thee cases. Case 1. Let k = 0. Conside the v 1 v v 3 v 4 path P 4. Clealy, f(v 1 ) = f(v 4 ) = 1 and f(v ) = f(v 3 ) = 1 define a maximum ISTDF of P 4 with weight 0. Case. Let k 1. Conside the path P k+4 on vetices v 1,..., v k+4, espectively, in which v 1 and v k+4 ae the leaves. Let T be a tee obtained fom P k+4 by adding two leaves to each vetex v i, fo all 3 i k +. The condition f(n(v)) 0, fo all ISTDF f and v V (T ), follows that all suppot vetices must be assigned 1 unde f. This shows that f(v ) =... = f(v k+3 ) = 1 and f(v) = 1, fo v v,..., v k+3, is a maximum ISTDF of T with weight k. Case 3. Let k 1. Let T i be a tee obtained fom a the path P 3 on vetices v i1, v i and v i3 by adding two leaves l 1 ij and l ij to v ij, fo j = 1,, 3. Let the tee T be the disjoint union of T 1,..., T k by adding a path on the vetices v 1,..., v k. Then f : V (T ) 1, 1} defined by 1 if v = v ij, l i1 f(v) =, l i3, 1 if v v ij, l i1, l i3 is a maximum ISTDF of T with weight k. We bound the ISTDN of a tee fom below by consideing its leaves and suppot vetices and chaacteize all tees attaining this bound. Fo this pupose, we intoduce some notation. The set of leaves and suppot vetices of a tee T ae denoted by L = L(T ) and S = S(T ), espectively. Conside L v as the set of all leaves adjacent to the suppot vetex v, and T as the subgaph of T induced by the set of suppot vetices. The following lemma will be useful. Lemma 4.. If T is a tee, then thee exists an ISTDF of T of the weight γst(t 0 ) that assigns to at least li leaves of the suppot vetex v i the value 1, whee l i is the numbe of leaves adjacent to v i. Poof. Let f : V 1, 1} be an ISTDF of weight γst(t 0 ). Suppose that thee exists a suppot vetex v i which is adjacent to at most li 1 vetices in P. Then, f(n(v i )) deg(v i )+( li 1). Let u be a leaf adjacent to v i with f(u) = 1. Define f : V 1, 1} by f 1 if v = u, (v) = f(v) if v u. Then f is an ISTDF of T with weight γst(t 0 )+, which is a contadiction. Theefoe, evey suppot vetex v i is adjacent to at least li vetices in P. Conside the suppot vetex v 1 and the vetices u 1,..., u l 1 in L v 1. Without loss of geneality, we can assume that u 1,..., u l 1 have the value 1 unde f. Since f is a maximum ISTDF then this statement holds fo the othe suppot vetices, as well.

8 454 D.A. Mojdeh and B. Samadi Now we define Ω to be the family of all tees T satisfying the following conditions: (a) fo any suppot vetex w we have L w o T is isomophic to the path P ; (b) (T ) 1; (b 1 ) if (T ) = 1, then all of vetices of T ae leaves o suppot vetices and L w is even fo all suppot vetex w; (b ) if (T ) = 0, then (i) T is isomophic to the sta K 1,n 1 o (ii) each suppot vetex is adjacent to just one vetex in V \ (L S), evey vetex in V \ (L S) has at least one neighbo in S and L w is even fo all suppot vetices w. We ae now in a position to pesent the following lowe bound. Theoem 4.3. Let T be a tee of ode n with the set of suppot vetices S = v 1,..., v s }. Then ( γst(t 0 l1 ls ) ) n , whee l i is the numbe of leaves adjacent to v i. Moeove, the equality holds if and only if T Ω. Poof. Let f : V 1, 1} be a function which assigns 1 to li leaves of v i and 1 to all emaining vetices. Then, it is easy to see that f is an ISTDF of T. Theefoe ( γst(t 0 l1 ls ) ) f(v ) = n Let T be a tee fo which the equality holds. So, we may assume that the above function f is a maximum ISTDF of T. We fist show that T satisfies (a). Without loss of geneality, we assume that T is not isomophic to the path P. Suppose that thee exists a suppot vetex v k adjacent to just one leaf u. Thus, s. Then the function g : V 1, 1} which assigns 1 to u and li leaves of v i, i k, and 1 to all othe vetices is an ISTDF of T with weight f(v ) + 1, which is a contadiction. Theefoe, all suppot vetices ae adjacent to at least two leaves. That the tee T satisfies (b), may be seen as follows. Let (T ). Then thee ae thee vetices v i 1, v i and v i+1 on a path as a subgaph of T. Let u be a leaf adjacent to v i with f(u) = 1. Since v i is adjacent to the vetices v i 1 and v i+1 with f(v i 1 ) = f(v i+1 ) = 1, then g : V 1, 1} defined by 1 if v = u, g(v) = f(v) if v u is an ISTDF of T with weight f(v ) +, a contadiction. Thus, (T ) 1. We now distinguish two cases depending on (T ) = 0 o 1. Case 1. (T ) = 1. Suppose to the contay that thee exists a vetex v in V \ (S L) adjacent to a suppot vetex v k with degee one in T. Then it must be assigned 1 unde f. Let u be a leaf adjacent to v k with f(u) = 1. We define a function h : V 1, 1} by 1 if v = u, h(v) = f(v) if v u.

9 On the invese signed total domination numbe in gaphs 455 Since vv k is an edge of T, then h is an ISTDF of T with weight f(v ) +, which is a contadiction. Theefoe all of vetices of T ae leaves o suppot vetices. This implies that S = v 1, v } and v 1 v is an edge of T. Now let L v1 be odd. Then the function h : V 1, 1} that assigns 1 to l1 + 1 leaves of v 1, l leaves of v and 1 to all emaining vetices is an ISTDF of T with weight f(v ) +, a contadiction. A simila agument shows that L v is even, as well. Case. (T ) = 0. If T is isomophic to the sta K 1,n 1, then (b ) holds. Othewise, s. Suppose to the contay that thee exists a suppot vetex w adjacent to at least two vetices u 1, u V \ (L S) and u is a leaf adjacent to w with f(u) = 1. Since f(u 1 ) = f(u ) = 1, then : V 1, 1} defined by (v) = 1 if v = u, f(v) if v u is an ISTDF of T with weight f(v ) +, which is a contadiction. Moeove, if u is a vetex in V \ (L S) with no neighbo in S then the function : V 1, 1} fo which (u) = 1 and (v) = f(v) fo all emaining vetices is an ISTDF of T with weight f(v ) +, this contadicts the fact that f is a maximum ISTDF of T. Finally, the poof of the fact that L w is even fo all suppot vetex w is simila to that of Case 1. These two cases imply that T satisfies (b 1 ) and (b ). Convesely, suppose that T Ω and f : V 1, 1} is an ISTDF of T with weight ( f(v ) = γst(t 0 l1 ls ) ) > n (4.1) By Lemma 4., we may assume that f assigns 1 to at least li leaves ui 1,..., u i l i of the suppot vetex v i, fo 1 i s. The inequality (4.1) shows that thee exists a vetex u V \ s i=1 ui 1,..., u i } such that f(u) = 1. Since all suppot vetices l i must be assigned 1 unde f, then u L vi o u is a vetex in V \ (L S) adjacent to a suppot vetex v i, fo some 1 i s. It is not had to see that this contadicts the fact that T belongs to Ω and f(n(v i )) 0, fo all 1 i s. This completes the poof. Acknowledgements The authos ae gateful to the anonymous efeees fo thei valuable comments and suggestions. REFERENCES [1] M. Atapou, S. Noouzian, S.M. Sheikholeslami, L. Volkmann, Bounds on the invese signed total domination numbes in gaphs, Opuscula Math. 36 (016), [] M.A. Henning, Signed total domination in gaphs, Discete Math. 78 (004),

10 456 D.A. Mojdeh and B. Samadi [3] S.M. Hosseini Moghaddam, D.A. Mojdeh, B. Samadi, L. Volkmann, New bounds on the signed total domination numbe of gaphs, Discuss. Math. Gaph Theoy 36 (016), [4] Z. Huang, Z. Feng, H. Xing, Invese signed total domination numbes of some kinds of gaphs, Infomation Computing and Applications Communications in Compute and Infomation Science ICICA 01, Pat II, CCIS 308 (01), [5] V. Kulli, On n-total domination numbe in gaphs, Gaph Theoy, Combinatoics, Algoithms and Applications, SIAM, Philadelphia, 1991, [6] D.A. Mojdeh, A. Sayed Khalkhali, H. Abdollahzadeh, Y. Zhao, Total k-distance domination citical gaphs, Tansaction on Combinatoics 5 (016) 3, 1 9. [7] P. Tuán, On an extemal poblem in gaph theoy, Math. Fiz. Lapok 48 (1941), [8] C. Wang, The negative decision numbe in gaphs, Austalas. J. Combin. 41 (008), [9] C. Wang, Lowe negative decision numbe in a gaph, J. Appl. Math. Comput. 34 (010), [10] H. Wang, E. Shan, Signed total -independence in gaphs, Utilitas Math. 74 (007), [11] D.B. West, Intoduction to Gaph Theoy, nd ed., Pentice Hall, 001. [1] B. Zelinka, Signed total domination numbe of a gaph, Czechoslovak Math. J. 51 (001), 5 9. D.A. Mojdeh damojdeh@umz.ac.i Univesity of Mazandaan Depatment of Mathematics Babolsa, Ian B. Samadi samadibabak6@gmail.com Univesity of Mazandaan Depatment of Mathematics Babolsa, Ian Received: July 1, 016. Revised: Decembe 8, 016. Accepted: Decembe 10, 016.

SUFFICIENT CONDITIONS FOR MAXIMALLY EDGE-CONNECTED AND SUPER-EDGE-CONNECTED GRAPHS DEPENDING ON THE CLIQUE NUMBER

Discussiones Mathematicae Gaph Theoy 39 (019) 567 573 doi:10.7151/dmgt.096 SUFFICIENT CONDITIONS FOR MAXIMALLY EDGE-CONNECTED AND SUPER-EDGE-CONNECTED GRAPHS DEPENDING ON THE CLIQUE NUMBER Lutz Volkmann