Discussiones Mathematicae Graph Theory 37 (2017) 989–1004 doi:10.7151/dmgt.1983 TWIN MINUS TOTAL DOMINATION NUMBERS IN DIRECTED GRAPHS Nasrin Dehgardi Department of Mathematics and Computer Science Sirjan University of Technology Sirjan, I.R. Iran e-mail: n.dehgardi@sirjantech.ac.ir and Maryam Atapour Department of Mathematics Faculty of Basic Sciences University of Bonab Bonab, I.R. Iran e-mail: m.atapour@bonabu.ac.ir Abstract Let D = (V, A) be a finite simple directed graph (shortly, digraph). A function f : V −→ {−1, 0, 1} is called a twin minus total dominating function (TMTDF) if f (N − (v)) ≥ 1 and f (N + (v)) ≥ 1 for each vertex v ∈ V . ∗ The twin minus total domination number of D is γmt (D) = min{w(f ) | f is a TMTDF of D}. In this paper, we initiate the study of twin minus total domination numbers in digraphs and we present some lower bounds for ∗ γmt (D) in terms of the order, size and maximum and minimum in-degrees and out-degrees. In addition, we determine the twin minus total domination numbers of some classes of digraphs. Keywords: twin minus total dominating function, twin minus total domination number, directed graph. 2010 Mathematics Subject Classification: 05C69. 990 N. Dehgardi and M. Atapour 1. Introduction In this paper, D is a finite simple directed graph with vertex set V (D) and arc set A(D) (briefly, V and A). A digraph without directed cycles of length 2 is + an oriented digraph. We write d+ D (v) = d (v) for the out-degree of a vertex v − − and dD (v) = d (v) for its in-degree. The minimum and maximum in-degree and minimum and maximum out-degree of D are denoted by δ − (D) = δ − , ∆− (D) = ∆− , δ + (D) = δ + and ∆+ (D) = ∆+ , respectively. If (u, v) is an arc of D, then we say that v is an out-neighbor of u and u is an in-neighbor of v, and we also say that − u dominates v or v is dominated by u. The sets N − (v) = ND (v) = {x | (x, v) ∈ + + A(D)} and N (v) = ND (v) = {x | (v, x) ∈ A(D)} are called the in-neighborhood − [v] = N − [v] = N − (v) ∪ {v} and out-neighborhood of the vertex v. Likewise, ND + − + + and ND [v] = N [v] = N (v)∪{v}. For S ⊆ V (D), we define N − (S) = ND (S) = S S − + + − − − + v∈S N (v), N [S] = ND [S] = N (S)∪S and v∈S N (v), N (S) = ND (S) = + N + [S] = ND [S] = N + (S) ∪ S. If X ⊆ V (D) and v ∈ V (D), then A(X, v) is the set of arcs from X to v. We denote by A(X, Y ) the set of arcs from a subset X to a subset Y . The notation D−1 is used for the digraph obtained from D by reversing the arcs of D. With any digraph D, we can associate a graph G with the same vertex set simply by replacing each arc by an edge with the same vertices. This graph is the underlying graph of D, denoted G(D). The complete digraph of order n, Kn∗ , is a digraph D such that (u, v), (v, u) ∈ A(D) for any two distinct vertices u,P v ∈ V (D). For a real-valued function f : V (D)P −→ R the weight of f is w(f ) = v∈V f (v), and for S ⊆ V , we define f (S) = v∈S f (v), so w(f ) = f (V ). Consult [13] for the notation and terminology which are not defined here. A signed total dominating function (abbreviated STDF) of D is a function f : V → {−1, 1} such that f (N − (v)) ≥ 1 for every v ∈ V . The signed total domination number of a digraph D is γst (D) = min{w(f ) | f is a STDF of D}. A γst (D)-function is a STDF of D of weight γst (D). The signed total domination number of a digraph was introduced by Sheikholeslami [12]. Recently, Atapour et al. [1] studied the twin signed total domination numbers in digraphs. A signed total dominating function of a digraph D is called a twin signed total dominating function (briefly, TSTDF) if it is also a signed total dominating function of D−1 , i.e., f (N + (v)) ≥ 1 for every v ∈ V . The twin signed total ∗ (D) = min{w(f ) | f is a TSTDF of D}. domination number of a digraph D is γst + Let D be digraph with min{δ (D), δ − (D)} ≥ 1. A minus total dominating function (abbreviated MTDF) of D is a function f : V → {−1, 0, 1} such that f (N − (v)) ≥ 1 for every v ∈ V . The minus total domination number for a digraph D is γmt (D) = min{w(f ) | f is a MTDF of D}. Twin Minus Total Domination Numbers in Directed Graphs 991 A γmt (D)-function is a MTDF of D of weight γmt (D). The minus total domination number of a digraph was introduced by Li et al. [10]. We define a twin minus total dominating function of D as a minus total dominating function of both D and D−1 , i.e., f (N − (v)) ≥ 1 and f (N + (v)) ≥ 1 for every ∗ (D) = v ∈ V . The twin minus total domination number for a digraph D is γmt − + min{w(f ) | f is a TMTDF of D}. As the assumption δ (D), δ (D) ≥ 1 is nec∗ (D), all digraphs involved essary, we always assume that when we discuss γmt − + satisfy δ (D) ≥ 1 and δ (D) ≥ 1. Let G be a graph with vertex V and edge set E. For every vertex v ∈ V , the open neighborhood N (v) is the set {u ∈ V | uv ∈ E} and the closed neighborhood of v is N [v] = N (v) ∪ {v}. A minus total dominating function of G, introduced by Harris et al. [7], is a function f : V → {−1, 0, 1} such that f (N (v)) ≥ 1 for every v ∈ V . The minus total domination number of G, denoted by γmt (G), is the minimum weight of a minus total dominating function on G. The minus total domination number in graphs and its related parameters was studied by several authors, for example [8, 9, 11, 15]. For any function f : V (D) → {−1, 0, 1}, on a digraph D, we define P = Pf = {v ∈ V | f (v) = 1}, Z = Zf = {v ∈ V | f (v) = 0} and M = Mf = {v ∈ V | f (v) = −1}. Since every TMTDF of D is a MTDF on both D and D−1 and since the constant function 1 is a TMTDF of D, we have (1) ∗ max{γmt (D), γmt (D−1 )} ≤ γmt (D) ≤ |V (D)|. Since every TSTDF of a digraph D is a TMTDF, we have ∗ ∗ γmt (D) ≤ γst (D). (2) In this paper, we initiate the study of the twin minus total domination number in digraphs and we present some lower bounds on this parameter. 2. Basic Properties In this section, we present basic properties of the twin minus total domination ∗ (D) ≤ |V (D)| for any digraph D. The next proposition number. By (1), γmt provides conditions to establish the equality. ∗ (D) = n if and only if Proposition 1. Let D be a digraph of order n. Then γmt every vertex has either an out-neighbor with in-degree at most 1 or an in-neighbor with out-degree at most 1. Proof. The sufficiency is clear. Thus, we verify the necessity of the condition. Assume to the contrary that there exists a vertex v ∈ V (D) such that d− (u) ≥ 2 992 N. Dehgardi and M. Atapour for each u ∈ N + (v) and d+ (w) ≥ 2 for each w ∈ N − (v). Define f : V (D) → {−1, 0, 1} by f (v) = 0 and f (x) = 1 for x ∈ V (D) \ {v}. Obviously, f is a twin minus total dominating function of D of weight less than n, a contradiction. This completes the proof. The next result is an immediate consequence of Proposition 1. → − → ∗ (− Corollary 2. If C n is the directed cycle on n vertices, then γmt C n ) = n. Now we show that the twin minus total domination and also the twin signed total domination number of digraphs can be arbitrarily small. Theorem 3. For any positive integer k ≥ 2, there exists a digraph D such that ∗ γmt (D) ≤ 6k − 4k 2 . Proof. Let k ≥ 2 be an integer and D be a digraph obtained from a complete ∗ ) = {u , u | 1 ≤ i ≤ k} by adding digraph of order 2k with vertex set V (K2k i1 i2 the set {vij , wij | 1 ≤ i ≤ k and 1 ≤ j ≤ 2k − 2} of new vertices and the set {(vij , ui1 ), (ui1 , wij ), (wij , ui2 ), (ui2 , vij ) | 1 ≤ i ≤ k, 1 ≤ j ≤ 2k − 2} of new arcs. It is easy to see that the function f : V (D) → {−1, 0, 1} defined by f (x) = 1 for x ∈ {ui1 , ui2 | 1 ≤ i ≤ k} and f (x) = −1 otherwise, is a TMTDF of ∗ (D) ≤ 6k − 4k 2 . D and so γmt The function defined in the proof of Theorem 3 is also a TSTDF of D and so ∗ (D) ≤ 6k − 4k 2 . Then the twin signed total domination number of digraphs γst can be arbitrarily small. ∗ (D) ≥ max{γ (D), γ (D −1 )}. Now we show As we observed in (1), γmt mt mt ∗ that the difference γmt (D) − max{γmt (D), γmt (D−1 )} can be arbitrarily large. Theorem 4. For every positive integer k ≥ 3, there exists a digraph D such that ∗ γmt (D) − max{γmt (D), γmt (D−1 )} ≥ k. Proof. Let k ≥ 3 be an integer and D be a digraph obtained from the di→ − rected cycle C k = (v1 , . . . , vk ) by adding new vertices ui , 1 ≤ i ≤ 2k, and arcs {(vi , ui ), (ui , uk+i ), (uk+i , vi ) | 1 ≤ i ≤ k}. Then the order of D is n = 3k. Obvi∗ (D) = n. On ously, D ∼ = D−1 and so, γmt (D) = γmt (D−1 ). By Proposition 1, γmt the other hand, it is easy to verify that the function f : V (D) → {−1, 0, 1} defined by f (x) = 1 for x ∈ {ui , vi | 1 ≤ i ≤ k} and f (x) = 0 otherwise, is a MTDF of ∗ (D) − max{γ (D), γ (D −1 )} ≥ k D and so γmt (D) ≤ 2k. This implies that γmt mt mt and the proof is complete. Twin Minus Total Domination Numbers in Directed Graphs 993 A tournament is a digraph in which for every pair u and v of different vertices, either (u, v) ∈ A(D) or (v, u) ∈ A(D), but not both. Next we determine the exact value of the twin minus total domination number for a particular type of tournaments. Let n = 2r + 1 for some positive integer r. We define the circulant tournament CT(n) with n vertices as follows. The vertex set of CT(n) is V (CT(n)) = {u0 , u1 , . . . , un−1 } and for each i, the arcs go from ui to the vertices ui+1 , . . . , ui+r where the sum being taken modulo n. The proof of the next result can be found in [10]. Proposition 5. For n ≥ 3, γmt (CT(n)) = 3. ∗ (CT(n)) = γ (CT(n)) The next proposition shows that γmt mt Proposition 6. Let n ≥ 3 and n = 2r + 1 where r is a positive integer. Then ∗ (CT(n)) = γ (CT(n)). γmt mt ∗ (CT(n)) ≥ 3. On the other hand, Proof. By (1) and Proposition 5, we have γmt the function f : V (CT(n)) → {−1, 0, 1} defined by f (u0 ) = f (ur ) = f (u2r ) = 1 and f (x) = 0 otherwise, is TMTDF of CT(n) of weight 3. This completes the proof. ∗ (D) ≥ γ ∗ (D). Next we show that γ ∗ (D) − γ ∗ (D) As we observed in (2), γst mt st mt can be arbitrarily large. The proof of the following proposition can be found in [1]. Proposition 7. Let D be a digraph of order n with δ + (D), δ − (D) ≥ 1. Then ∗ (D) = n if and only if every vertex has either an out-neighbor with indegree at γst most 2 or an in-neighbor with outdegree at most 2. Theorem 8. For every positive integer k, there exists a digraph D such that ∗ ∗ γst (D) − γmt (D) ≥ 2k. Proof. Let k ≥ 1 be an integer and for 1 ≤ j ≤ k, let Dj be a circulant tournament CT(5) with vertex set {uij | 1 ≤ i ≤ 5}. Let D be a digraph obtained from the union of Dj ’s by adding the set {(u31 , u32 ), (u32 , u33 ), . . . , (u3k−1 , u3k ), ∗ (D) = 5k. (u3k , u31 )} of new arcs. Then the order of D is 5k. By Proposition 7, γst On the other hand, it is easy to see that the function f : V (D) → {−1, 0, 1} defined by f (u1j ) = f (u3j ) = f (u5j ) = 1 and f (x) = 0 otherwise, is a TMTDF ∗ (D) ≤ 3k. It follows that γ ∗ (D) − γ ∗ (D) ≥ 2k and the proof is of D and so γmt st mt complete. 994 N. Dehgardi and M. Atapour 3. ∗ (D) Lower Bounds on γmt ∗ (D) in terms of the order, In this section we present some lower bounds for γmt size, the maximum and minimum in-degrees and out-degrees of D. We begin with some results on the minus total domination number of a digraph. Observation 9. Let f be any γmt (D)-function of a digraph D of order n. Then 1. n = |M | + |P | + |Z|. 2. w(f ) = |P | − |M |. Theorem 10. Let f be an MTDF on a digraph D of order n. If ∆+ = ∆+ (D) and δ + = δ + (D), then (a) (∆+ − 1)|P | ≥ (δ + + 1)|M | + |Z|. (b) (∆+ + δ + )|P | + δ + |Z| ≥ (δ + + 1)n. (c) δ + w(f ) ≥ (δ + − ∆+ )|P | + n. (d) w(f ) ≥ 2δ + −∆+ +1 n ∆+ −δ + + |P |. Proof. (a) It follows from Observation 9 (part 1) that X X X f (x) = d+ (v)f (v) |P | + |M | + |Z| = n ≤ = X v∈P v∈V x∈N − (v) X + d (v) − v∈M v∈V + d (v) ≤ ∆+ |P | − δ + |M |. This inequality chain yields to the desired bound in (a). (b) Observation 9 (part 1) implies that |M | = n − |P | − |Z|. Using this identity and part (a), we arrive at (b). (c) According to Observation 9 and part (b), we obtain part (c) as follows: w(f ) = 2|P | − n + |Z| and δ + w(f ) = δ + (2|P | − n + |Z|) = (∆+ + δ + )|P | + (δ + − ∆+ )|P | − δ + n + δ + |Z| ≥ (δ + − ∆+ )|P | − δ + n + (δ + + 1)n = (δ + − ∆+ )|P | + n. (d) The inequality chain in the proof of part (a) and Observation 9 (part 1) show that n ≤ ∆+ |P ∪ Z| − δ + (n − |P ∪ Z|) = (∆+ − δ + )|P ∪ Z| − δ + n and so |P ∪ Z| ≥ δ+ + 1 n. ∆+ − δ + Twin Minus Total Domination Numbers in Directed Graphs 995 Using this inequality and Observation 9, we obtain w(f ) = |P | − n + |P ∪ Z| ≥ δ+ + 1 2δ + − ∆+ + 1 n − n + |P | = n + |P |. ∆+ − δ + ∆+ − δ + This is the bound in part (d), and the proof is complete. Corollary 11. Let D be a digraph of order n, minimum out-degree δ + and maximum out-degree ∆+ . If δ + < ∆+ , then γmt (D) ≥ 2δ + − ∆+ + 2 n. ∆+ Proof. Multiplying both sides of the inequality in Theorem 10 (d) by (∆+ − δ + ) and adding the resulting inequality to the inequality in Theorem 10 (c), we obtain the desired lower bound. Since δ + (D−1 ) = δ − (D) and ∆+ (D−1 ) = ∆− (D) for any digraph D, Corollary 11 implies the following corollary. Corollary 12. Let D be a digraph of order n, minimum in-degree δ − and maximum in-degree ∆− . If δ − < ∆− , then γmt (D−1 ) ≥ 2δ − − ∆− + 2 n. ∆− The next corollary is a consequence of (1) and Corollaries 11 and 12. Corollary 13. Let D be a digraph of order n, minimum in-degree δ − , maximum in-degree ∆− , minimum out-degree δ + and maximum out-degree ∆+ . If δ − < ∆− and δ + < ∆+ , then  +  2δ − ∆+ + 2 2δ − − ∆− + 2 ∗ γmt (D) ≥ max n, n . ∆+ ∆− ∗ (D) ≥ 6 − n with Proposition 14. Let D be a digraph of order n ≥ 3. Then γmt → − equality if and only if D = C 3 . ∗ (D)-function. Since D is a simple digraph, we have |P ∩ Proof. Let f be a γmt + − (N (v) ∪ N (v))| ≥ 2 for every v ∈ P . This implies that |P | ≥ 3 and so ∗ (D) ≥ 6 − n. |M | ≤ n − 3. Hence γmt ∗ (D) = 6−n and f be a γ ∗ (D)-function. Let now D be a digraph such that γmt mt Then |P | = 3, |M | = n−3 and |Z| = 0. Also since D is a simple digraph, for every v ∈ P , |N − (v) ∩ P | = |N + (v) ∩ P | = 1 and so |N − (v) ∩ M | = |N + (v) ∩ M | = 0. → − → − It follows that M = ∅ and so D = C 3 . On the other hand, if D = C 3 , then ∗ (D) = 3 = 6 − n. γmt 996 N. Dehgardi and M. Atapour Proposition 15. If D is a digraph with maximum out-degree ∆+ (D) ≤ 4, then ∗ (D) ≥ 0. γmt Proof. Let f be a twin minus total dominating function for which ω(f ) = ∗ (D). If M = ∅, then the result follows. Assume that M 6= ∅. Since γmt f (N − (v)) ≥ 1, for each v ∈ M , we have |A(P, v)| ≥ 1. It follows that |A(P, M )| ≥ |M |. On the other hand, since f (N + (v)) ≥ 1 for each v ∈ P , we have |A(v, P )| ≥ + (v) ≤ ∆+ (D) ≤ 4, it follows that |A(v, M )| + 1. Since |A(v, P )| + |A(v, M )| = dP |A(v, M )| ≤ 1. Hence, we have |A(P, M )| = v∈P |A(v, M )| ≤ |P |. Combining ∗ (D) = |P | − |M | ≥ 0. these, we have |M | ≤ |P |, and so γmt The condition ∆− (D) ≤ 4, in the Proposition 15, implies the following proposition. Proposition 16. If D is a digraph with maximum in-degree ∆− (D) ≤ 4, then ∗ (D) ≥ 0. γmt A twin minus total dominating function f of D is called minimal if there exists no twin minus total dominating function f ′ of D such that f ′ 6= f and f ′ (v) ≤ f (v) for every v ∈ V (D). Proposition 17. A twin minus total dominating function f on a digraph D is minimal if and only if for every vertex v ∈ V with f (v) ≥ 0, there exists a vertex u ∈ N + (v) with f (N − (u)) = 1 or there exists a vertex w ∈ N − (v) with f (N + (w)) = 1. Proof. Let f be a minimal twin minus total dominating function and assume that there is a vertex v ∈ V with f (v) ≥ 0, f (N + (u)) > 1 for every u ∈ N − (v) and f (N − (w)) > 1 for every w ∈ N + (v). Define a new function g : V → {−1, 0, 1} by g(v) = f (v) − 1 and g(x) = f (x) for all x 6= v. Then for all u ∈ N + (v), g(N + (u)) = f (N + (u)) − 1 ≥ 1, g(N − (u)) = f (N − (u)) ≥ 1, for all w ∈ N − (v), g(N + (w)) = f (N + (w)) ≥ 1, g(N − (w)) = f (N − (w)) − 1 ≥ 1 and for z∈ / N + (v) ∪ N − (v), g(N + (z)) = f (N + (z)) ≥ 1 and g(N − (z)) = f (N − (z)) ≥ 1. This implies that g is a twin minus total dominating function of D, contradiction to the minimality of f . Conversely, let f be a twin minus total dominating function such that for all v ∈ V with f (v) ≥ 0, there exists a vertex u ∈ N + (v) with f (N − (u)) = 1 or there exists a vertex w ∈ N − (v) with f (N + (w)) = 1. Assume that f is not minimal, i.e., there is a twin minus total dominating function g such that g 6= f and g(x) ≤ f (x) for all x ∈ V . Then there is at least one v ∈ V with g(v) < f (v). It follows that f (v) ≥ 0, and by assumption, there exists a vertex u ∈ N + (v) with f (N − (u)) = 1 or there exists a vertex w ∈ N − (v) with f (N + (w)) = 1. Since g(x) ≤ f (x) for all x ∈ V and g(v) < f (v), we have g(N − (u)) < f (N − (u)) = 1 or g(N + (w)) < f (N + (w)) = 1. This contradicts the fact that g is a twin minus Twin Minus Total Domination Numbers in Directed Graphs 997 total dominating function. Hence f is a minimal twin minus total dominating function and this completes the proof. Theorem 18. Let D be a digraph of order n and size m. Then ∗ γmt (D) ≥ 2n − m, and this bound is sharp. ∗ (D)-function. For any v ∈ M , we have |A(v, P )| ≥ 1 Proof. Let f be a γmt and |A(P, v)| ≥ 1 which implies that |A(M, P )| ≥ |M | and |A(P, M )| ≥ |M |. Also for any v ∈ Z, we have |A(v, P )| ≥ 1 and |A(P, v)| ≥ 1 which implies that |A(Z, P )| ≥ |Z| and |A(P, Z)| ≥ |Z|. On the other hand, if x ∈ P , then it follows from f (N + (x)) ≥ 1 that |A(x, P )| ≥ |A(x, M )| + 1 implying that |A(P, P )| ≥ |A(P, M )| + |P | ≥ |M | + |P |. Therefore, (3) m ≥ |A(M, P )| + |A(P, M )| + |A(Z, P )| + |A(P, Z)| + |A(P, P )| ≥ 2|M | + 2|Z| + |M | + |P | = 2|M | + 2|Z| + n −|Z| = n + 2|M | + |Z|. Hence, we have ∗ (D) = w(f ) = |P | − |M | = n − 2|M | − |Z| ≥ 2n − m. γmt → − To prove the sharpness, suppose that C t is a directed cycle of order t ≥ 3. → − Let D be a digraph obtained from C t by adding the set {u1 , . . . , uk | k ≥ 1} → − of new vertices and arcs from vertices of C t to k new vertices and from k new → − − vertices to vertices of C t such that d+ D (v) = dD (v) = 1 for every new vertex v. Then the order of D is n = t + k and the size of D is m = t + 2k. So 2n − m = t. → − Now define f : V (D) → {−1, 0, 1} which assigns f (x) = 1 for x ∈ V ( C t ) and f (x) = 0 otherwise. Obviously, f is a TMTDF of D and ω(f ) = t. This completes the proof. A set S ⊆ V (G) is a 2-packing if for each pair of vertices x, y ∈ S, N [x] ∩ N [y] = ∅. The 2-packing number ρ(G) is the cardinality of a maximum 2-packing. Proposition 19. Let G be a graph of order n with minimum degree δ ≥ 2 and let D be an orientation of G such that δ + (D) ≥ 1, δ − (D) ≥ 1. Then ∗ γmt (D) ≥ ρ(G)(δ + 1) − n. ∗ (D)-function. Since Proof. Let S be a maximum 2-packing of G and f be a γmt + − f (N (v)) ≥ 1 and f (N (v)) ≥ 1, we have f (NG (v)) = f (N + (v))+f (N − (v)) ≥ 2 for each v ∈ S. This implies that 998 N. Dehgardi and M. Atapour ∗ (D) = γmt X f (NG (v)) + v∈S ≥ |S| + X f (v) v∈V (G)−NG (S) X v∈V (G)−NG (S) (−1) ≥ |S| − (n − |S|δ) = ρ(G)(δ + 1) − n, and the proof is complete. The next theorem presents a lower bound on twin signed total domination numbers in a digraph in terms of its order. Theorem 20. Let D be a digraph of order n. Then √ ∗ γst (D) ≥ 1 + 1 + 4n − n. ∗ (D)-function. For every v ∈ M , since f (N + (v)) ≥ 1, we Proof. Let f be a γst have |A(v, P )| ≥ 1 and thus |A(M, P )| ≥ |M |. If x ∈ P , then it follows from f (N + (x)) ≥ 1 that |A(x, P )| ≥ |A(x, M )| + 1. This implies that |A(P, P )| ≥ |A(P, M )| + |P | ≥ |M | + (n − |M |) = n. On the other hand, |A(P, P )| ≤ |P |(|P | − 1). It follows that |P |(|P | − 1) ≥ n and so |P |2 − |P | − n ≥ 0. This implies that √ 1 + 4n + 1 , |P | ≥ 2 and thus we obtain ∗ γst (D) = 2|P | − n ≥ 1 + √ 4n + 1 − n. The next theorem presents a lower bound on twin signed total domination numbers in a bipartite digraph in terms of its order. Theorem 21. Let D be a bipartite digraph of order n. Then √ ∗ γst (D) ≥ 2 2n − n. ∗ (D)-function. In view of the proof of Theorem 20, |A(P, P )| Proof. Let f be a γst ≥ n. Since the subdigraph induced by P √is bipartite, we have |A(P, P )| ≤ |P |2 /2. It follows that |P |2 /2 ≥ n and so |P | ≥ 2n. This implies that √ ∗ γst (D) = 2|P | − n ≥ 2 2n − n. Twin Minus Total Domination Numbers in Directed Graphs 999 Next, we present lower bounds on twin minus total domination numbers in digraphs in terms of their orders. Theorem 22. Let D be a digraph of order n. Then ∗ γmt (D) ≥ 1 + √ 1 + 4n − n. ∗ (D)-function. If Z = ∅, then f is a TSTDF on D and Proof. Let f be a γmt √ ∗ ∗ (D) ≥ 1 + 1 + 4n − n. Suppose Z 6= ∅. by Theorem 20, γmt (D) = w(f ) ≥ γst Let n1 = n − |Z| and D1 be a subdigraph of D induced by the set V (D) − Z. ∗ (D) = w(f ) ≥ γ ∗ (D ) ≥ Then f |V (D1 ) is a TSTDF on D1 and by Theorem 20, γmt √ st 1 √ 1+ 1 + 4n1 −n1 . Now we can easily see that the function g(x) = 1+ 1 + 4x−x is a non increasing function √ for any integer x ≥√2 and so g(n1 ) ≥ g(n). This ∗ (D) ≥ 1 + 1 + 4n − n ≥ 1 + 1 + 4n − n. implies that γmt 1 1 Theorem 23. Let D be a bipartite digraph of order n. Then √ ∗ γmt (D) ≥ 2 2n − n. ∗ (D)-function. If Z = ∅, then f is a TSTDF on D and Proof. Let f be a γmt √ ∗ (D) = w(f ) ≥ γ ∗ (D) ≥ 2 2n − n. Suppose Z 6= ∅. Let by Theorem 21, γmt st n1 = n − |Z| and D1 be a subdigraph of D induced by the set V (D) − Z. Then ∗ (D) = w(f ) ≥ γ ∗ (D ) ≥ f |V (D1 ) is a TSTDF on D1 and by Theorem 21, γmt 1 st √ √ 2 2n1 − n1 . Now we can easily see that the function g(x) = 2 2x − x is a non increasing function for any√integer x ≥ 2 and so g(n1 ) ≥ g(n). This implies that √ ∗ (D) ≥ 2 2n − n ≥ 2 2n − n. γmt 1 1 The associated digraph D(G) of a graph G is the digraph obtained in such a way that each edge e of G is replaced by two oppositely oriented arcs with + − (v) = ND(G) (v) = NG (v) for each v ∈ the same end vertices as e. Since ND(G) V (G) = V (D(G)), the following useful observation is valid. ∗ (D(G)) Observation 24. If D(G) is the associated digraph of a graph G, then γmt = γmt (G). Theorems 22, 23 and Observation 24 lead to the next well-known result. √ Corollary 25 [14]. If G is a graph of order n, then √ γmt (G) ≥ 1 + 4n + 1 − n. If G is a bipartite graph of order n, then γmt (G) ≥ 2 2n − n. Xing et al. [14] have presented examples with equality in the two inequalities of Corollary 25. The associated digraphs of these examples show that Theorems 22 and 23 are both sharp. 1000 N. Dehgardi and M. Atapour 4. Twin Minus Total Domination in Oriented Graphs Let G be the complete bipartite graph K4,4 with bipartite sets V1 = {v1 , v2 , v3 , v4 } and V2 = {u1 , u2 , u3 , u4 }. Let D1 be a 2-regular oriented graph of G and D2 be an orientation of G such that A(D2 ) = {(vi , ui ), (uj , vr ) | 1 ≤ i, j, r ≤ 4 and j 6= r}. ∗ (D ) = 4 and γ ∗ (D ) = 8. Thus two distinct orientations It is easy to see that γmt 2 1 mt of a graph can have distinct twin minus total domination numbers. Motivated by this observation, we define lower orientable twin minus total domination number dom∗mt (G) and upper orientable twin minus total domination number Dom∗mt (G) of a graph G with δ(G) ≥ 2 as follows: ∗ dom∗mt (G) = min{γmt (D) | D is an orientation of G}, and ∗ Dom∗mt (G) = max{γmt (D) | D is an orientation of G}. Corresponding concepts have been defined and studied for orientable domination (out-domination) [6], twin domination number [5], twin signed domination number [3], twin signed total domination number [1], twin minus domination number [2] and twin signed Roman domination number [4]. Note that the definitions are well-defined because every graph G with δ(G) ≥ 2, has an orientation D such that δ + (D), δ − (D) ≥ 1. Since for any orientation ∗ (D) ≤ γ ∗ (D), we have D of a graph G, γmt st (4) dom∗mt (G) ≤ dom∗st (G) Proposition 26. For any graph G of order n with δ(G) ≥ 2, γmt (G) ≤ dom∗mt (G). ∗ (D)-function. Then Proof. Let D be an orientation of G and let f be a γmt − + + (v)) + f (ND (v)) for each v ∈ V . Since f (ND (v)) ≥ 1 and f (NG (v)) = f (ND − f (ND (v)) ≥ 1, we have f (NG (v)) ≥ 2 for each v ∈ V , and so f is a MTDF of G. Therefore γmt (G) ≤ w(f ) = dom∗mt (G) as desired. The proof of the next result is straightforward and therefore omitted. Proposition 27. Let G be a graph of order n and v ∈ V (G). If deg(u) ≤ 3, for ∗ (D)-function f , some u ∈ NG (v), then for any orientation D of G and any γmt we have f (v) ≥ 0. Proposition 28. For n ≥ 3, dom∗mt (Cn ) = n. Proof. If D is an orientation of Cn with δ + (D) ≥ 1 and δ − (D) ≥ 1, then obviously D is a directed cycle and the result follows from Corollary 2. Twin Minus Total Domination Numbers in Directed Graphs 1001 We now proceed to determine the lower orientable twin minus total domination numbers of several classes of graphs including complete graphs, complete bipartite graphs and wheels. Lemma 29. For n ≥ 3, dom∗mt (Kn ) ≥ 3. Proof. The result is immediate for n = 3. Let n ≥ 4, D be an orientation of Kn ∗ (D)-function. Assume that v ∈ P . Since f (N + (v)) ≥ 1 and and let f be a γmt D − f (ND (v)) ≥ 1, we have + − dom∗mt (Kn ) = w(f ) = f (ND (v)) + f (ND (v)) + f (v) ≥ 3, as desired. Theorem 30. For n ≥ 3, dom∗mt (Kn ) = 3. Proof. The result is immediate for n = 3, so assume n ≥ 4. Let D1 be an orientation of Kn−3 with vertex set V (Kn−3 ) = {ui | 1 ≤ i ≤ n − 3}. Suppose that D is obtained from D1 by adding the set {v1 , v2 , v3 } of new vertices and the set {(v1 , v2 ), (v2 , v3 ), (v3 , v1 ), (v1 , ui ), (v2 , ui ), (ui , v3 ) | 1 ≤ i ≤ n − 3} of new arcs. Then D is an orientation of Kn . It is easy to see that the function f : V (D) → {−1, 0, 1} defined by f (v1 ) = f (v2 ) = f (v3 ) = 1 and f (x) = 0 otherwise, is a TMTDF of D of weight 3. This implies that dom∗mt (Kn ) ≤ w(f ) = 3. Now the result follows from Lemma 29. Lemma 31. For m, n ≥ 2, dom∗mt (Km,n ) ≥ 4. Proof. Let V (Km,n ) = X ∪ Y . Let D be an orientation of Km,n and let f ∗ (D)-function. Assume that v ∈ X and u ∈ Y . Since f (N + (v)) ≥ 1 be a γmt D − + − and f (ND (v)) ≥ 1, and since Y = f (ND (v)) ∪ f (ND (v)), f (Y ) ≥ 2. Similarly, f (X) ≥ 2. It follows that dom∗mt (Km,n ) = w(f ) = f (X) + f (Y ) ≥ 4, as desired. Theorem 32. For m, n ≥ 2, dom∗mt (Km,n ) = 4. Proof. Let D1 be an orientation of Km−2,n−2 with vertex set V (Km−2,n−2 ) = {ui , vj | 1 ≤ i ≤ m − 2, 1 ≤ j ≤ n − 2} and suppose that D is obtained from D1 by adding the set {w1 , w2 , w3 , w4 } of new vertices and the set {(w1 , w2 ), (w2 , w3 ), (w3 , w4 ), (w4 , w1 ), (w1 , ui ), (ui , w3 ), (w2 , vj ), (vj , w4 ) | 1 ≤ i ≤ m − 2, 1 ≤ j ≤ n − 2} of new arcs. Then D is an orientation of Km,n . It is easy to see that the function f : V (D) → {−1, 0, 1} defined by f (wk ) = 1 for 1 ≤ k ≤ 4 and f (x) = 0 otherwise, is a TMTDF of D of weight 4. This implies that dom∗mt (Km,n ) ≤ w(f ) = 4. Now the result follows from Lemma 31. 1002 N. Dehgardi and M. Atapour The wheel, Wn , is a graph with vertex set {w, v0 , . . . , vn−1 } and edge set {wvi , vi vi+1 | 0 ≤ i ≤ n − 1} where the indices are taken modulo n. Next we determine the lower orientable twin minus total domination number of wheels. Lemma 33. For n ≥ 3,   n+3     2  n ≡ 2 (mod 4), ∗ n+1 n ≡ 0 (mod 4), dommt (Wn ) ≥ 2    n+2  otherwise. 2 ∗ (D) = dom∗ (W ) and f be a Proof. Let D be an orientation of Wn with γmt n mt ∗ γmt (D)-function. It follows from Proposition 27 that f (vi ) ≥ 0 for each 0 ≤ i ≤ n − 1. Also since for each 0 ≤ i ≤ n − 1, d+ (vi ) = 1 or d− (vi ) = 1, then f (w) ≥ 0. If f (w) = 0, then f (vi ) = 1 for each 0 ≤ i ≤ n − 1 and so   n+3     2  n ≡ 2 (mod 4), ∗ n+1 n ≡ 0 (mod 4), dommt (Wn ) = w(f ) = n ≥ 2    n+2  otherwise. 2 Let f (w) = 1. Suppose that vi ∈ P for some 0 ≤ i ≤ n − 1. Since d+ (vi ) = 1 or d− (vi ) = 1, then f (vi−1 ) = 1 or f (vi+1 ) = 1. Also it is easy to see that f (vi ) + f (vi+1 ) + f (vi+2 ) + f (vi+3 ) ≥ 2, for each 0 ≤ i ≤ n − 1. It follows that |{vi , vi+1 , vi+2 , vi+3 } ∩ P | ≥ 2, for each 0 ≤ i ≤ n − 1. Summing the above inequalities for each 0 ≤ i ≤ n − 1, we have |P | ≥ |Z| + 3 when n ≡ 2 (mod 4), |P | ≥ |Z| + 1 when n ≡ 0 (mod 4) and |P | ≥ |Z| + 2 otherwise. It follows that   n+3     2  n ≡ 2 (mod 4), ∗ n+1 n ≡ 0 (mod 4), dommt (Wn ) = w(f ) = |P | ≥ 2    n+2  otherwise. 2 Theorem 34. For n ≥ 3,   n+3     2  n ≡ 2 (mod 4), ∗ n+1 n ≡ 0 (mod 4), dommt (Wn ) = 2    n+2  otherwise. 2 Proof. Let D be an orientation of Wn such that  A(D) = (vi , vi+1 ), (w, v4j ), (v4j+1 , w), (v4j+2 , w), (w, v4j+3 ) | 0 ≤ i ≤ n − 1,   0 ≤ j ≤ n4 . It is easy to verify that the function  n  f : V (D) → {−1, 0, 1} defined by f (x) = 1 for x ∈ w, v4j , v4j+1 | 0 ≤ j ≤ 4 and f (x) = 0 otherwise, is a TMTDF of Twin Minus Total Domination Numbers in Directed Graphs 1003       D of weight n+3 when n ≡ 2 (mod 4), n+1 when n ≡ 0 (mod 4) and n+2 2 2 2 otherwise. This implies that   n+3     2  n ≡ 2 (mod 4), ∗ n+1 n ≡ 0 (mod 4), dommt (Wn ) ≤ w(f ) = 2    n+2  otherwise. 2 Now the result follows from Lemma 33. References [1] M. Atapour, A. Bodaghli and S.M. Sheikholeslami, Twin signed total domination numbers in directed graphs, Ars Combin., to appear. 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Shan, Upper minus total domination in small-degree regular graphs, Discrete Math. 307 (2007) 2453–2463. doi:10.1016/j.disc.2006.11.011 Received 18 March 2016 Revised 22 August 2016 Accepted 22 August 2016