EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 14, No. 2, 2021, 451-470 ISSN 1307-5543 – ejpam.com Published by New York Business Global Forcing Subsets for γ∗tpw -sets in Graphs Cris L. Armada1 1 Mathematics Department, College of Arts and Sciences, Cebu Normal University, Cebu City, Philippines 6000 Abstract. In this paper, the lower and upper bounds of the forcing total dr-power domination number of any graph are determined. Total dr-power domination number of some special graphs such as complete graphs, star, fan and wheel graphs are shown. Moreover, the forcing total dr-power domination number of these graphs, together with paths and cycles, are determined. 2020 Mathematics Subject Classifications: 05C38, 05C69 Key Words and Phrases: Forcing, total, dr-power domination, paths, cycles 1. Introduction Let G = (V,E) be a graph representing the electrical power system, where a vertex represents an electrical node and an edge represents a transmission line joining two electrical nodes. In order to monitor the power system, some measurement devices must be placed at selected locations so that all the state variables of the system can be measured. A Phase Measurement Unit (PMU) is a measurement device placed on a vertex and has the ability to measure the state of the vertex and the edges connected to the vertex. The vertices and edges that are measured by PMU’s are said to be observed. In this study, it is necessary that each vertex with PMU is adjacent to another vertex with PMU also. But because of the high cost value of a PMU, it is desirable to minimize their number while maintaining the ability to monitor the entire power system. All graphs considered in this study are simple, undirected and without loops or multiple edges. Let G = (V (G), E(G)) be a graph and v ∈ V (G). The open neighborhood of v in G is the set N(v) = {u ∈ V (G) : uv ∈ E(G)} and the closed neighborhood of v is the set N [v] = N(v) ∪ {v}. For X ⊆ V (G), the open neighborhood of X is the set N(X) = ∪v∈XNG(v) and its closed neighborhood is the set N [X] = N(X) ∪X. A set S ⊆ V (G) is a dominating set (resp. total dominating set) of G if N [S] = V (G) (resp. N(S) = V (G)). The domination number γ(G) (resp. total domination number DOI: https://doi.org/10.29020/nybg.ejpam.v14i2.3914 Email addresses: armadac@cnu.edu.ph / cris.armada@g.msuiit.edu.ph (C. Armada) http://www.ejpam.com 451 c© 2021 EJPAM All rights reserved. C. Armada / Eur. J. Pure Appl. Math, 14 (2) (2021), 451-470 452 γt(G)) of G is the minimum cardinality of a dominating set (resp. total dominating set). If S is a dominating set (resp. a total dominating set) with |S| = γ(G) (resp. |S| = γt(G)), then we call S a γ-set (resp. a γt-set) of G. Let G = (V,E) be a simple graph. Let P ⊆ V (G). An edge e = uv of G is directly observed by P if u ∈ P or v ∈ P . A vertex u of G is directly observed if u is incident to a directly observed edge. An edge e′ = xy is remotely observed by P if x, y /∈ P and x, y are directly observed vertices or at least one of x and y is incident to k edges where k − 1 of these edges are directly observed by P . Clearly, k is a positive integer, k > 1, and k is not constant for any pair of vertices x and y. A non-directly observed vertex u of G which is incident to a remotely observed edge is called remotely observed vertex. Let OP V (G) be the set of all directly and remotely observed vertices and OP E(G) be the set of all directly and remotely observed edges. Then P ⊆ V (G) is a dr-power dominating set (dr-pds) of G if OP V (G) = V (G) and OP E(G) = E(G). The minimum cardinality of a dr-power dominating set is called the dr-power domination number of G and is denoted by γ∗pw(G). A subset P of V (G) with cardinality γ∗pw(G) is called a γ∗pw-set of G. A dr-power dominating set D is said to be a total dr-power dominating set(tdr-pds) if the induced subgraph 〈D〉 has no isolated vertex. The minimum cardinality of a total dr-power dominating set (tdr-pds) is called the total dr-power domination number of G and is denoted by γ∗tpw (G). A subset T of V (G) with cardinality γ∗tpw (G) is called a γ∗tpw -set of G. Moreover, there exists a connected graph G such that 2 ≤ γ∗tpw (G) ≤ γt(G). Let S be a γ∗tpw -set of a graph G. A subset D of S is said to be a forcing subset for S if S is the unique γ∗tpw -set containing D. The forcing total dr-power domination number of S is given by fγ∗tpw (S) = min{|D| : D is a forcing subset for S}. The forcing total dr-power domination number of G is given by fγ∗tpw (G) = min{fγ∗tpw (S) : S is a γ∗tpw -set of G}. The join of two graphs G and H, denoted by G+H is the graph with vertex set V (G+H) = V (G) ∪ V (H) and edge set E(G+H) = E(G) ∪ E(H) ∪ {uv : u ∈ V (G), v ∈ V (H)}. The total domination is studied by Amos [1]. Chartrand et al. [5] investigated the relation between forcing and domination concepts and defined "forcing domination number". Canoy, et al studied the following concepts: total dr-power domination [6], forcing domination number of graphs under some binary operations [7], forcing total domination number and forcing connected domination number under the lexicographic product of graphs [8], forcing independent domination number of a graph [4], and A-differential of graphs [3]. Also, Armada [2] studied the forcing total dr-power domination of graphs under some binary operations. C. Armada / Eur. J. Pure Appl. Math, 14 (2) (2021), 451-470 453 Illustration 1.1. Consider the cycle graph C5 = [u1, u2, u3, u4, u5, u1]. Let P ⊆ V (C5). Pick u2, u3 ∈ P . Then u1u2, u2u3 and u3u4 are directly observed edges in C5. Clearly, u1, u2, u3 and u4 are incident to a directly observed edge, and so, u1, u2, u3 and u4 are directly observed vertices. The edges u1u5 and u4u5 are remotely observed edges since u1, u4, u5 /∈ P and there are k = 2 incident edges to the vertices u1 and u4 such that k − 1 = 2− 1 = 1 edge is directly observed by P which are u1u2 and u3u4. Since u5 is incident to a remotely observed edge u1u5 or u4u5, then u5 is a remotely observed vertex. Clearly, OP V (C5) = V (C5) and OP E(C5) = E(C5), that is, P is a dr-power dominating set of C5. Since the induced subgraph 〈P 〉 has no isolated vertex, P is a total dr-power dominating set of C5. Note that for any connected graph G, γ∗tpw (G) ≥ 2 and since |P | = 2, P is a γ∗tpw -set of C5 and γ∗tpw (C5) = 2. Clearly, any pair of adjacent vertices in C5 is a γ∗tpw -set of C5, that is, S1 = {u1, u2}, S2 = P = {u2, u3}, S3 = {u3, u4}, S4 = {u4, u5}, S5 = {u5, u1}, are the only γ∗tpw -sets of C5. Clearly, for all i = 1, 2, . . . , 5, no subset {ui} is contained in a unique γ∗tpw -set Sj for all j = 1, 2, . . . , 5 and so, fγ∗tpw (Sj) 6= 1. Therefore, for all j = 1, 2, . . . , 5, fγ∗tpw (Sj) = |Sj | = 2 = fγ∗tpw (C5). 2. Known Results This section contains known results involving dr-power domination, total domination and total dr-power domination numbers of a graph G that are useful in proving the main results of this study. Remark 2.1. [6] For any graph G without isolated vertices, γ∗pw(G) ≤ γ∗tpw (G) ≤ γt(G). Theorem 2.2. [6] Let n be a positive integer with n ≥ 5. Then γ∗tpw (Pn) =  2n 5 , n ≡ 0(mod 5) 2n−2 5 , n ≡ 1(mod 5) 2n+1 5 , n ≡ 2(mod 5) 2n+4 5 , n ≡ 3(mod 5) 2n+2 5 , n ≡ 4(mod 5) Theorem 2.3. [6] Let n be a positive integer with n ≥ 5. Then γ∗tpw (Cn) =  2n 5 , n ≡ 0(mod 5) 2n+3 5 , n ≡ 1(mod 5) 2n+6 5 , n ≡ 2(mod 5) 2n+4 5 , n ≡ 3(mod 5) 2n+2 5 , n ≡ 4(mod 5) C. Armada / Eur. J. Pure Appl. Math, 14 (2) (2021), 451-470 454 Proposition 2.4. [1] The total domination number of a cycle Cn or a path Pn on n ≥ 3 vertices is given by γt(Cn) = γt(Pn) =  n 2 , n ≡ 0(mod 4), n+2 2 , n ≡ 2(mod 4), n+1 2 , otherwise. Theorem 2.5. [6] Let G and H be any graphs. Then P ⊆ V (G+H) is a total dr-power dominating set of G+H if and only if it satisfies one of the following conditions: (i) P ⊆ V (G) and is a total dominating set, provided that G is a graph with no isolated vertex; (ii) P ⊆ V (H) and is a total dominating set, provided that H is a graph with no isolated vertex; or (iii) P = P1 ∪ P2, where ∅ 6= P1 ⊆ V (G) and ∅ 6= P2 ⊆ V (G). Corollary 2.6. [6] Let G and H be any graphs. Then γ∗tpw (G+H) = 2. 3. Main Results This section contains the lower and upper bounds of fγ∗tpw (G) and the forcing total dr-power domination number of some special graphs such as path, cycle, complete graph, fan, star and wheel graphs. Theorem 3.1. Let G be a graph. Then (i) fγ∗tpw (G) = 0 if and only if G has a unique γ∗tpw -set. (ii) fγ∗tpw (G) = 1 if and only if G has at least two γ∗tpw -sets and there exists a vertex v which is contained in exactly one γ∗tpw -set of G. Proof. (i) Suppose that fγ∗tpw (G) = 0. It follows that ∅ is the forcing subset for a γ∗tpw -set, say P , in G. Suppose that there is another γ∗tpw -set of G, say R. Note that ∅ is also a subset for R, a contradiction since ∅ is a forcing subset for P . Thus, G has a unique γ∗tpw -set. Conversely, if G has a unique γ∗tpw -set, say Q. Clearly, ∅ is a forcing subset for Q. Consequently, |∅| = 0 = fγ∗tpw (Q) = fγ∗tpw (G). (ii) Suppose that fγ∗tpw (G) = 1. Hence, G has at least two γ∗tpw -sets by part (i) and there exists a γ∗tpw -set, say P , and v ∈ P such that {v} is a forcing subset for P and fγ∗tpw (P ) = |{v}| = 1, that is, there exists a vertex which is contained in exactly one γ∗tpw -set of G. Conversely, if G has at least two γ∗tpw -sets, then fγ∗tpw (G) > 0 by part (i). By assumption, there exists a vertex, say x, which is contained in exactly one γ∗tpw -set of G, say T , that is, {x} is a forcing subset for T . Therefore, fγ∗tpw (T ) = |{x}| = 1 = fγ∗tpw (G). The next result is a direct consequence of Theorem 3.1 and definition of forcing total dr-power domination. C. Armada / Eur. J. Pure Appl. Math, 14 (2) (2021), 451-470 455 Corollary 3.2. Let G be a connected graph. Then 0 ≤ fγ∗tpw (G) ≤ γ∗tpw (G). Theorem 3.3. Let G be a nontrivial graph. Then fγ∗tpw (G) = γ∗tpw (G) if and only if for every γ∗tpw -set P of G and for each v ∈ P , there exists u ∈ V (G)\P such that [P\{v}]∪{u} is a γ∗tpw -set of G. Proof. Suppose that fγ∗tpw (G) = γ∗tpw (G). Let P be a γ∗tpw -set of G such that fγ∗tpw (G) = |P | = γ∗tpw (G), that is, P is the only forcing subset for P . Let v ∈ P . Since P\{v} is not a forcing subset for P , there exists a u ∈ V (G)\P such that [P\{v}] ∪ {u} is a γ∗tpw -set of G. Conversely, suppose that every γ∗tpw -set P ′ of G satisfies the given condition. Let P be a γ∗tpw -set of G such that fγ∗tpw (G) = fγ∗tpw (P ). Moreover, suppose that P has a forcing subset R with |R| < |P |, that is, P = R ∪ S, where S = {u ∈ P : u /∈ R}. Pick u ∈ S. By assumption, there exists v ∈ V (G)\P such that [P\{u}] ∪ {v} = Q is a γ∗tpw -set of G. Thus, Q = R ∪ T , where T = [S\{u}] ∪ {v}, that is, Q is a γ∗tpw -set containing R, a contradiction. Thus, |R| = |P | and P is the only forcing subset for P . Therefore, fγ∗tpw (G) = |P | = γ∗tpw (G). Theorem 3.4. Let n be a positive integer with n ≥ 5. Then fγ∗tpw (Pn) =  0, n = 7 or n ≡ 1(mod 5) 2, n ≡ 3(mod 5) 1, otherwise. Proof. Let the path Pn = [u1, u2, . . . , un]. Note that deg(u1) = 1 = deg(un) and deg(ui) = 2 for all i = 2, 3, . . . , n − 1. By definition of total dr-power dominating set, a choosen vertex in a γ∗tpw -set, say D, of Pn must always have an adjacent vertex in D, say ui and ui+1 and by Theorem 2.2, the choosen vertex ui+1 must be of at most distance 4 to the next choosen vertex in D since if the next vertex to be choosen is of distance 5, say u1, u2 ∈ D and choose u7 to be the next vertex, then u3 and u6 are directly observed vertices while u4 and u5 are remotely observed vertices but the edge u4u5 is neither directly nor remotely observed edge which is a contradiction. Also, the starting vertex of a γ∗tpw -set D, of Pn must be u1, u2 or u3 since if it starts with u4, then u3 is a directly observed vertex while u2 becomes a remotely observed vertex but the vertex u1 is neither directly nor remotely observed vertex which is a contradiction. Now, consider the following cases: Case 1: Suppose that n = 7. By Theorem 2.2, γ∗tpw (P7) = 2(7)+1 5 = 3. Clearly, S = {u3, u4, u5} is the only γ∗tpw -set of P7 since u2 and u6 are the directly observed vertices while u1 and u7 are the remotely observed vertices, that is, OS V (P7) = V (P7), OS E(P7) = E(P7) and the induced subgraph 〈S〉 has no isolated vertex. By Theorem 3.1(i), fγ∗tpw (P7) = 0. C. Armada / Eur. J. Pure Appl. Math, 14 (2) (2021), 451-470 456 Case 2: Suppose that n ≡ 1(mod 5). By Theorem 2.2, γ∗tpw (Pn) = 2n−2 5 . Let n = 6. Then γ∗tpw (P6) = 2(6)−2 5 = 2. Clearly, S = {u3, u4} is the only γ∗tpw -set of P6 since u2 and u5 are the directly observed vertices while u1 and u6 are the remotely observed vertices, that is, OS V (P6) = V (P6) and OS E(P6) = E(P6) and the induced subgraph 〈S〉 has no isolated vertex. By Theorem 3.1(i), fγ∗tpw (P6) = 0. Now, suppose that n > 6. Let p = n−1 5 and j = 0, 1, 2, . . . , p− 1, p. Group the vertices of Pn into p+ 1 disjoint subsets Rj R0 = {u1} R1 = {u2, u3, u4, u5, u6} R2 = {u7, u8, u9, u10, u11} ... Rp−1 = {un−9, un−8, un−7, un−6, un−5} Rp = {un−4, un−3, un−2, un−1, un} Let i = 2, 7, 12, . . . , n− 4. For every induced subgraph 〈ui, ui+1, ui+2, ui+3, ui+4〉, the ver- tices ui+1, ui+2 form a total dr-power dominating set since ui and ui+3 are directly observed vertices while u1 and ui+4 are remotely observed vertices for all i = 2, 7, 12, . . . , n− 9, n− 4. Let the set R = {ui+1, ui+2 : i = 2, 7, 12, . . . , n− 9, n− 4} = {u3, u4, u8, u9, . . . , un−8, un−7, un−3, un−2} where |R| = 2p = 2n−2 5 , OR V (Pn) = V (Pn), OR E(Pn) = E(Pn), and the induced subgraph 〈R〉 has no isolated vertex. By Theorem 2.2, R is a γ∗tpw -set of Pn. Note that the set R contains pairs of adjacent vertices in each Rj ’s except in R0 and the distance between the last choosen vertex in Rj and the first choosen vertex in Rj+1 is always 4. Let T be a γ∗tpw -set of Pn different from R. Consider the following subcases: Subcase 1: Let u2, u3 ∈ T . Then the next vertex to be chosen must be of distance 4 from u3, that is, the vertex u7, together with u8, must be in T . Thus, ui, ui+1 ∈ T for all i = 2, 7, . . . , n− 9, n− 4. It follows that T = {u2, u3, u7, u8, . . . , un−9, un−8, un−4, un−3} and |T | = |R|. Since un−3 is the last vertex in T , the vertex un−2 is a directly observed vertex and un−1 is a remotely observed vertex. Hence, un is neither a directly or a remotely observed vertex. Thus, OT V (Pn) 6= V (Pn), a contradiction, that is, T is not a γ∗tpw -set of Pn. Hence, it is not possible to start with the vertex u2 to form a γ∗tpw -set T . Similarly, it is not possible to start with the vertex u1. Subcase 2: Suppose that u3, u4 ∈ T . Now, replace u8 ∈ R by u7 to form T . Then the next vertex to be chosen must be u8, that is, the vertex ui, ui+1 must be in T for all i = 7, 12, . . . , n − 9, n − 4. Then T = {u3, u4, u7, u8, u13, u14, . . . , un−4, un−3} and |T | = |R|. Since un−3 is the last ver- tex in T , by previous subcase, T is not a γ∗tpw -set of Pn. Since u8 is arbitrarily replaced from R, we cannot replace the vertex ui+1 in R, where i = 7, 12, . . . , n− 9, n− 4 to form C. Armada / Eur. J. Pure Appl. Math, 14 (2) (2021), 451-470 457 another γ∗tpw -set of Pn. Thus, either of the subcases, it is not possible to form another γ∗tpw -set T of Pn which is different from R. Therefore, R is a unique γ∗tpw -set of Pn. By Theorem 3.1(i), fγ∗tpw (Pn) = 0. Case 3: Suppose that n 6= 7 and n ≡ 2(mod 5). By Theorem 2.2, γ∗tpw (Pn) = 2n+1 5 . Let n = 12. Then γ∗tpw (P12) = 2(12)+1 5 = 5. Clearly, S1 = {u3, u4, u5, u9, u10} and S2 = {u3, u4, u8, u9, u10} are the only γ∗tpw -sets of P12. Since u5 ∈ S1 and u5 /∈ S2, by Theorem 3.1(ii), fγ∗tpw (P12) = 1. Now, suppose that n > 12. Let p = n−2 5 and j = 0, 1, 2, . . . , p− 1, p. Group the vertices of Pn into p+ 1 disjoint subsets Rj R0 = {u1, u2} R1 = {u3, u4, u5, u6, u7} R2 = {u8, u9, u10, u11, u12} R3 = {u13, u14, u15, u16, u17} ... Rp−1 = {un−9, un−8, un−7, un−6, un−5} Rp = {un−4, un−3, un−2, un−1, un} Let i = 3, 8, 13, . . . , n − 4. For every induced subgraph 〈ui, ui+1, ui+2, ui+3, ui+4〉, the vertices u3, ui+1, ui+2 form a total dr-power dominating set since u2, ui and ui+3 are directly observed vertices while u1 and ui+4 are remotely observed vertices for all i = 3, 8, 13, . . . , n− 9, n− 4. Let the set R = {u3, ui+1, ui+2 : i = 3, 8, 13, . . . , n− 9, n− 4} = {u3, u4, u5, u9, u10, u14, u15, . . . , un−8, un−7, un−3, un−2} where |R| = 2p + 1 = 2 ( n−2 5 ) + 1 = 2n+1 5 , OR V (Pn) = V (Pn), OR E(Pn) = E(Pn), and the induced subgraph 〈R〉 has no isolated vertex. By Theorem 2.2, R is a γ∗tpw -set of Pn. Let m+ 1 be the number of γ∗tpw -sets of Pn where m is a positive integer. Let k = 1, 2, . . . ,m and Tk be a γ∗tpw -set of Pn different from R. Note that in forming R, there are three vertices in R1 such that the induced subgraph is a graph P3 and two adjacent vertices in the other Rj ’s with j > 1. Thus, Tk can be formed by getting 3 vertices in any one of the Rj ’s where j > 1 such that the induced subgraph is a graph P3 and two adjacent vertices in the other Rl’s where l 6= j and l 6= 0. Consider the following subcases. Subcase 1: Choose 3 vertices in R2 to form another γ∗tpw set, say T1, that is, replaced u5 ∈ R1 in R by u8 ∈ R2. It follows that T1 = {u3, u4, u8, u9, u10, u14, u15, . . . , un−8, un−7, un−3, un−2}, where |T1| = |R|, OT1 V (Pn) = V (Pn), OT1 E (Pn) = E(Pn), and the induced subgraph 〈T1〉 has no isolated vertex, that is, T1 is a γ∗tpw -set of Pn. Clearly, u5 /∈ T1. Subcase 2: Choose 3 vertices in R3 to form another γ∗tpw set, say T2, C. Armada / Eur. J. Pure Appl. Math, 14 (2) (2021), 451-470 458 that is, replaced u10 ∈ R2 in T1 by u13 ∈ R3. It follows that T2 = {u3, u4, u8, u9, u13, u14, u15, u19, u20, . . . , un−8, un−7, un−3, un−2}, where |T2| = |R|, OT2 V (Pn) = V (Pn), OT2 E (Pn) = E(Pn), and the induced subgraph 〈T2〉 has no isolated vertex, that is, T2 is a γ∗tpw -set of Pn. Clearly, u5 /∈ T2. Continuing in this manner and in any subcase, u5 /∈ Tk for all k = 1, 2, . . . ,m, and so, the vertex u5 is contained in γ∗tpw -set R only. By Theorem 3.1(ii), fγ∗tpw (Pn) = 1. Case 4: Suppose that n ≡ 3(mod 5). By Theorem 2.2, γ∗tpw (Pn) = 2n+4 5 . Suppose that n = 8. Then γ∗tpw (P8) = 2(8)+4 5 = 4. Clearly, S1 = {u1, u2, u5, u6}, S2 = {u1, u2, u6, u7}, S3 = {u2, u3, u5, u6}, S4 = {u2, u3, u6, u7}, S5 = {u2, u3, u7, u8}, S6 = {u3, u4, u5, u6}, S7 = {u3, u4, u6, u7}, and S8 = {u3, u4, u7, u8} are the γ∗tpw -sets of P8. Clearly, for i = 1, 2, . . . , 8, no subset {ui} is contained in exactly one of the Sl’s, for l = 1, 2, . . . , 8, that is, fγ∗tpw (Sl) > 1. Clearly, {u1, u5} is forcing subset for S1 since {u1, u5} * Sl for all l 6= 1. Thus, fγ∗tpw (S1) = 2 = fγ∗tpw (P8). Now, suppose that n > 8. Since fγ∗tpw (P8) = 2, fγ∗tpw (Pn) ≥ 2. Let p = n−3 5 and j = 0, 1, 2, . . . , p− 1, p. Group the vertices of Pn into p+ 1 disjoint subsets Rj R0 = {u1, u2, u3} R1 = {u4, u5, u6, u7, u8} R2 = {u9, u10, u11, u12, u13} R3 = {u14, u15, u16, u17, u18} ... Rp−1 = {un−9, un−8, un−7, un−6, un−5} Rp = {un−4, un−3, un−2, un−1, un} Let i = 4, 9, 14, . . . , n − 4. For every induced subgraph 〈ui, ui+1, ui+2, ui+3, ui+4〉, the vertices u1, u2, ui+1, ui+2 form a total dr-power dominating set since u3, ui and ui+3 are directly observed vertices while ui+4 are remotely observed vertices for all i = 4, 9, 14, . . . , n− 9, n− 4. Let the set R = {u1, u2, ui+1, ui+2 : i = 4, 9, 14, . . . , n− 9, n− 4} = {u1, u2, u5, u6, u10, u11, u15, u16, . . . , un−8, un−7, un−3, un−2} where |R| = 2p + 2 = 2 ( n−3 5 ) + 2 = 2n+4 5 , OR V (Pn) = V (Pn), OR E(Pn) = E(Pn), and the induced subgraph 〈R〉 has no isolated vertex. By Theorem 2.2, R is a γ∗tpw -set of Pn. Let m+ 1 be the number of γ∗tpw -sets of Pn where m is a positive integer. Let k = 1, 2, . . . ,m and Tk be a γ∗tpw -set of Pn different from R. Consider the following subcases: Subcase 1: Tk, say T1 and T2, can be formed by replacing u1, u2 in R by either u2, u3 or u3, u4. It follows that T1 = {u2, u3, u5, u6, u10, u11, . . . , un−8, un−7, un−3, un−2} and the set T2 = {u3, u4, u5, u6, u10, u11, . . . , un−8, un−7, un−3, un−2} are γ∗tpw -sets of Pn. Clearly, C. Armada / Eur. J. Pure Appl. Math, 14 (2) (2021), 451-470 459 {u1, u5} * T1 and {u1, u5} * T2. Subcase 2: Tk, say T3, can be formed by replacing u5 in R by u7. Thus, the set T3 = {u1, u2, u6, u7, u10, u11, . . . , un−8, un−7, un−3, un−2} is a γ∗tpw -set of Pn. By the previous subcase, Tk can be formed by replacing u1, u2 in T3 by either u2, u3 or u3, u4. It follows that T4 = {u2, u3, u6, u7, u10, u11, . . . , un−8, un−7, un−3, un−2} and the set T5 = {u3, u4, u6, u7, u10, u11, . . . , un−8, un−7, un−3, un−2} are γ∗tpw -sets of Pn. Clearly, {u1, u5} * T3, {u1, u5} * T4 and {u1, u5} * T5. Subcase 3: Suppose that {u1, u5} is a subset in one of the Tk’s with k > 5, say T6, that is, u2 must be in T6 and either u4 or u6 is in T6. Suppose that u4 ∈ T6. Then the next pair to be choosen must be u9 and u10 since the distance between u5 and u9 is 4, that is, ui and ui+1 must be in T6. Thus, T6 = {u1, u2, u4, u5, u9, u10, u14, u15, . . . , un−4, un−3}. Since un−3 is the last vertex in T6, by subcase 1 of case 2, T6 is not a γ∗tpw -set of Pn. Now, suppose that u6 ∈ T6. Then the first four vertices u1, u2, u5, u6 in T6 are the same with R. Replace u10 ∈ R by u9 to form T6. Then the next vertex to be chosen must be u10, that is, the vertex ui, ui+1 must be in T6 for all i = 9, . . . , n − 9, n − 4. Then T6 = {u1, u2, u5, u6, u9, u10, u14, u15, . . . , un−4, un−3}. Since un−3 is the last vertex in T6, by subcase 1 of case 2, T6 is not a γ∗tpw -set of Pn. Since u10 is arbitrarily replaced from R, we cannot replace the vertex ui+1 in R, where i = 9, 14, . . . , n− 9, n− 4 to form another γ∗tpw -set of Pn. Hence, {u1, u5} is not a subset of Tk for all k > 5. Therefore, in any subcase, {u1, u5} * Tk for all k = 1, 2, . . . ,m, that is, {u1, u5} is a forcing subset for R. Thus, fγ∗tpw (R) = 2 = fγ∗tpw (Pn). Case 5: Suppose that n ≡ 4(mod 5). By Theorem 2.2, γ∗tpw (Pn) = 2n+2 5 . Let n = 9. Then γ∗tpw (P9) = 2(9)+2 5 = 4. Clearly, S1 = {u1, u2, u6, u7}, S2 = {u2, u3, u6, u7}, S3 = {u2, u3, u7, u8}, S4 = {u3, u4, u7, u8}, and S5 = {u3, u4, u8, u9} are the only γ∗tpw -sets of P9. Since u1 ∈ S1 and u1 /∈ Sl for l = 2, 3, 4, 5, by Theorem 3.1(ii), fγ∗tpw (P9) = 1. Now, suppose that n > 9. Let p = n−4 5 and j = 0, 1, 2, . . . , p− 1, p. Group the vertices of Pn into p+ 1 disjoint subsets Rj R0 = {u1, u2, u3, u4} R1 = {u5, u6, u7, u8, u9} R2 = {u10, u11, u12, u13, u14} R3 = {u15, u16, u17, u18, u13} ... Rp−1 = {un−9, un−8, un−7, un−6, un−5} Rp = {un−4, un−3, un−2, un−1, un} Let i = 5, 10, 15, . . . , n − 4. For every induced subgraph 〈ui, ui+1, ui+2, ui+3, ui+4〉, the vertices u1, u2, ui+1, ui+2 form a total dr-power dominating set since u3, ui and ui+3 C. Armada / Eur. J. Pure Appl. Math, 14 (2) (2021), 451-470 460 are directly observed vertices while u4 and ui+4 are remotely observed vertices for all i = 5, 10, 15, . . . , n− 9, n− 4. Let the set R = {u1, u2, ui+1, ui+2 : i = 5, 10, 15, . . . , n− 9, n− 4} = {u1, u2, u6, u7, u11, u12, u16, u17, . . . , un−8, un−7, un−3, un−2} where |R| = 2p + 2 = 2 ( n−4 5 ) + 2 = 2n+2 5 , OR V (Pn) = V (Pn), OR E(Pn) = E(Pn), and the induced subgraph 〈R〉 has no isolated vertex. By Theorem 2.2, R is a γ∗tpw -set of Pn. Let m+ 1 be the number of γ∗tpw -sets of Pn where m is a positive integer. Let k = 1, 2, . . . ,m and Tk be a γ∗tpw -set of Pn different from R. Consider the following subcases: Subcase 1: Tk, say T1 and T2, can be formed by replacing u1, u2 in R by either u2, u3 or u3, u4. It follows that T1 = {u2, u3, u6, u7, u11, u12, . . . , un−8, un−7, un−3, un−2} and the set T2 = {u3, u4, u6, u7, u11, u12, . . . , un−8, un−7, un−3, un−2} are γ∗tpw -sets of Pn. Clearly, u1 /∈ T1 and u1 /∈ T2. Subcase 2: Let k = 3 and let u1, u2 ∈ T3. Now, replace u6 ∈ R by u5 to form T3. Then the next vertex to be chosen must be u6, that is, the vertex ui, ui+1 must be in T3 for all i = 5, 10, 15, . . . , n− 9, n− 4. Then T3 = {u1, u2, u5, u6, u10, u11, . . . , un−4, un−3} such that |T3| = |R|. Since un−3 is the last vertex in T3, by subcase 1 of case 2, T3 is not a γ∗tpw -set of Pn. Since u6 is arbitrarily replaced from R, we cannot replace the vertex ui+1 in R, where i = 10, 15, . . . , n−9, n−4 to form another γ∗tpw -set of Pn. Thus, u1 /∈ Tk for all k ≥ 3. Therefore, in any subcase, u1 /∈ Tk for all k = 1, 2, . . . ,m and so, the vertex u1 is contained in γ∗tpw -set R only. By Theorem 3.1(ii), fγ∗tpw (Pn) = 1. Case 6: Suppose that n ≡ 0(mod 5). By Theorem 2.2, γ∗tpw (Pn) = 2n 5 . Let n = 5. Then γ∗tpw (P5) = 2(5) 5 = 2. Clearly, S1 = {u2, u3} and S2 = {u3, u4} are the only γ∗tpw -sets of P5. Since u2 ∈ S1 and u2 /∈ S2, by Theorem 3.1(ii), fγ∗tpw (P5) = 1. Now, suppose that n > 5. Let p = n 5 and j = 1, 2, . . . , p− 1, p. Group the vertices of Pn into p disjoint subsets Rj C. Armada / Eur. J. Pure Appl. Math, 14 (2) (2021), 451-470 461 R1 = {u1, u2, u3, u4, u5} R2 = {u6, u7, u8, u9, u10} R3 = {u11, u12, u13, u14, u15} ... Rp−1 = {un−9, un−8, un−7, un−6, un−5} Rp = {un−4, un−3, un−2, un−1, un} Let i = 1, 6, 11, . . . , n− 4. For every induced subgraph 〈ui, ui+1, ui+2, ui+3, ui+4〉, the vertices ui+1, ui+2 form a total dr-power dominating set since ui and ui+3 are directly observed vertices while ui+4 is a remotely observed vertex for all i = 1, 6, 11, . . . , n−9, n−4. Let the set R = {ui+1, ui+2 : i = 1, 6, 11, . . . , n− 9, n− 4} = {u2, u3, u7, u8, u12, u13, . . . , un−8, un−7, un−3, un−2} ,where |R| = 2p = 2n 5 , OR V (Pn) = V (Pn), OR E(Pn) = E(Pn), and the induced subgraph 〈R〉 has no isolated vertex. By Theorem 2.2, R is a γ∗tpw -set of Pn. Let m+ 1 be the number of γ∗tpw -sets of Pn where m is a positive integer. Let k = 1, 2, . . . ,m and Tk be a γ∗tpw -set of Pn different from R. Consider the following subcases: Subcase 1: Let k = 1. T1 can be formed by replacing u2 in R by u4. It follows that T1 = {u3, u4, u7, u8, u12, u13, . . . , un−8, un−7, un−3, un−2} is a γ∗tpw -set of Pn. Clearly, u2 /∈ T1. Subcase 2: Let k = 2 and let u2, u3 ∈ T2. Now, replace u7 ∈ R by u6 to form T2. Then the next vertex to be chosen must be u7, that is, the vertices ui, ui+1 must be in T2 for all i = 6, 11, 16, . . . , n− 9, n− 4. Then T2 = {u2, u3, u6, u7, u11, u12, . . . , un−4, un−3} such that |T2| = |R|. Since un−3 is the last vertex in T2, by subcase 1 of case 2, T3 is not a γ∗tpw -set of Pn. Since u7 is arbitrarily replaced from R, we cannot replace the vertex ui+1 in R, where i = 6, 11, 16, . . . , n− 9, n− 4 to form another γ∗tpw -set of Pn. Thus, it is not possible to start with vertex u2 to form Tk for all k ≥ 2, that is, u2 /∈ Tk for all k ≥ 2. Therefore, in any subcase, u2 /∈ Tk for all k = 1, 2, . . . ,m, and so, the vertex u2 is contained in γ∗tpw -set R only. By Theorem 3.1(ii), fγ∗tpw (Pn) = 1. C. Armada / Eur. J. Pure Appl. Math, 14 (2) (2021), 451-470 462 Theorem 3.5. Let n be a positive integer with n ≥ 5. Then fγ∗tpw (Cn) = { 4, n ≡ 2(mod 5) 2, otherwise. Proof. Let the cycle Cn = [u1, u2, . . . , un, u1]. Note that and deg(ui) = 2 for all i = 1, 2, 3, . . . , n− 1, n. By definition of total dr-power dominating set, a choosen vertex in a γ∗tpw -set, say D, of Cn must always have an adjacent vertex in D, say ui and ui+1 and by Theorem 2.3, the distance of the choosen vertex ui+1 must be of at most 4 to the next choosen vertex in D since if the next vertex to be choosen is of distance 5, say u1, u2 ∈ D and choose u7 to be the next vertex, then u3 and u6 are directly observed vertices while u4 and u5 are remotely observed vertices but the edge u4u5 is neither directly nor remotely observed edge which is a contradiction. Note that ui is contained in γ∗tpw -sets containing the pairs of sets {ui−1, ui} and {ui, ui+1} for all i = 1, 2, . . . n, and so, the set {ui} is not a forcing subset of any γ∗tpw -set of Cn, that is, fγ∗tpw (Cn) ≥ 2. Now, consider the following cases: Case 1: Suppose that n ≡ 2(mod 5). By Theorem 2.3, γ∗tpw (Cn) = 2n+6 5 . Suppose that n = 7. Then γ∗tpw (C7) = 2(7)+6 5 = 4. Clearly, S1 = {u1, u2, u3, u4}, S2 = {u1, u2, u4, u5}, S3 = {u1, u2, u5, u6}, S4 = {u1, u2, u6, u7}, S5 = {u2, u3, u4, u5}, S6 = {u2, u3, u5, u6}, S7 = {u2, u3, u6, u7}, S8 = {u3, u4, u5, u6}, S9 = {u3, u4, u6, u7}, S10 = {u4, u5, u6, u7}, S11 = {u7, u1, u2, u3}, S12 = {u7, u1, u3, u4}, S13 = {u7, u1, u4, u5} and S14 = {u7, u1, u5, u6} are the only γ∗tpw -sets of C7. Note that fγ∗tpw (C7) ≥ 2 and each pair of vertices in C7 is contained in more than one γ∗tpw -sets of C7, that is, fγ∗tpw (C7) ≥ 3. Note that each of the vertices u1, u2, u3, and u4 in S1 can be replaced by u5, u7, u5 and u7, respectively to form another γ∗tpw -sets of C7 which are S5, S12, S2, and S11, respectively, that is, fγ∗tpw (S1) = 4. Clearly, for all l = 1, 2, . . . , 14 and for every γ∗tpw -set Sl of C7 and for each ui ∈ Sl, there exists uj ∈ V (C7)\Sl and j 6= i such that [Sl\{ui}] ∪ {uj} is a γ∗tpw -set of C7. By Theorem 3.3, fγ∗tpw (C7) = γ∗tpw (C7) = 4. Now, suppose that n > 7. Since fγ∗tpw (C7) = 4, fγ∗tpw (Cn) ≥ 4. Let p = n−2 5 and j = 0, 1, 2, . . . , p− 1, p. Group the vertices of Cn into p+ 1 disjoint subsets Rj R0 = {u1, u2} R1 = {u3, u4, u5, u6, u7} R2 = {u8, u9, u10, u11, u12} R3 = {u13, u14, u15, u16, u17} ... Rp−1 = {un−9, un−8, un−7, un−6, un−5} Rp = {un−4, un−3, un−2, un−1, un} Let i = 3, 8, 13, . . . , n − 4. For every induced subgraph 〈ui, ui+1, ui+2, ui+3, ui+4〉, the vertices u1, u2, ui, ui+1 form a total dr-power dominating set since ui+2 and ui+4 are directly observed vertices while ui+3 is a remotely observed vertex for all i = 3, 8, 13, . . . , n−9, n−4. C. Armada / Eur. J. Pure Appl. Math, 14 (2) (2021), 451-470 463 Let the set R = {u1, u2, ui, ui+1 : i = 3, 8, 13, . . . , n− 9, n− 4} = {u1, u2, u3, u4, u8, u9, u13, u14, . . . , un−9, un−8, un−4, un−3} ,where |R| = 2p+ 2 = 2(n−2 5 ) + 2 = 2n+6 5 , OR V (Cn) = V (Cn), OR E(Cn) = E(Cn), and the induced subgraph 〈R〉 has no isolated vertex. By Theorem 2.3, R is a γ∗tpw -set of Cn. Let m+1 be the number of γ∗tpw -sets of Cn where m is a positive integer. Let k = 1, 2, . . . ,m+1 and Tk be a γ∗tpw -set of Cn and one of the Tk’s is equal to R. Note that in forming R, there are four vertices taken both from R0 and R1 such that the induced subgraph is a graph P4 and two adjacent vertices in the other Rj ’s where j > 1. Now, Tk can be formed by starting all the vertices of Sl for l = 1, 2, . . . , 14 and replacing u7 by un in S11, S12, S13, and S14, that is, Tk = Sl ∪H for some set H. If Tk starts with S11 = {un, u1, u2, u3} , say T1, note that S11 is the only set in the Sl’s that ends with u3 and its induced subgraph is a graph P4, then the next pairs of vertices must be choosen are u7, u8, u12, u13, . . . , un−5, un−4, that is, T1 = {un, u1, u2, u3, u7, u8, u12, u13, . . . , un−5, un−4} such that |T1| = |R|, OT1 V (Cn) = V (Cn), OT1 E (Cn) = E(Cn), and the induced subgraph 〈T1〉 has no isolated vertex, that is, T1 is a γ∗tpw -set of Cn. Note also that if Tk starts with S14 = {un, u1, u5, u6}, say T2, then the next pairs of vertices must be choosen are u7, u8, u12, u13, . . . , un−5, un−4 such that induced subgraph of {u5, u6, u7, u8} ⊆ T2 is a graph P4, that is, T2 = {un, u1, u5, u6, u7, u8, u12, u13, . . . , un−5, un−4} such that |T2| = |R|, OT2 V (Cn) = V (Cn), OT2 E (Cn) = E(Cn), and the induced subgraph 〈T2〉 has no isolated vertex, that is, T2 is a γ∗tpw -set of Cn. Clearly, T1\S11 = T2\S14. Now, since none of the other Sl’s ended with a unique vertex, the set of vertices in Tk\Sl must be contained in Tr for some r 6= k. Hence, Tk\Sl is not a forcing subset for Tk. Therefore, either the sets Sl or Tk must be the forcing subset for Tk for some l = 1, 2, . . . , 14 and for some k = 1, 2, . . . ,m+1. Now, if Tk starts with S1, then let k = 3 and {u1, u2, u3, u4} ⊆ T3 such that T3 is a γ∗tpw -set of Cn different from R. Replace u8 ∈ R by u7 to form T3. Then the next vertex to be chosen must be u8, that is, the vertex ui+4, ui must be in T3 for all i = 3, 8, 13, . . . , n − 9, n − 4. Then T3 = {u1, u2, u3, u4, u7, u8, u12, u13, . . . , un−5, un−4} such that |T3| = |R|. Since u1 ∈ T3 and un−4 is the last vertex in T3, the vertices un and un−3 are directly observed vertices while un−1 and un−2 are remotely observed vertices. Hence, by definition, the edge un−1un−2 is neither a directly or a remotely observed edge. Thus, OT3 E (Cn) 6= E(Cn), a contradiction, that is, T3 is not a γ∗tpw -set of Cn. Since u8 is arbitrarily replaced from R, we cannot replace the vertex ui in R, where i = 8, 13, . . . , n− 9, n− 4 to form another γ∗tpw -set of Cn. Therefore, only the γ∗tpw -set R starts with S1 and so, {u1, u2, u3, u4} * Tk for all Tk 6= R. Hence, {u1, u2, u3, u4} is a forcing subset for R. Therefore, fγ∗tpw (R) = 4 = fγ∗tpw (Cn). Case 2: Suppose that n ≡ 0(mod 5). By Theorem 2.3, γ∗tpw (Cn) = 2n 5 . Suppose that n = 5. Then γ∗tpw (C5) = 2(5) 5 = 2. Clearly, S1 = {u1, u2}, S2 = {u2, u3}, S3 = {u3, u4}, S4 = {u4, u5}, S5 = {u5, u1}, are the only γ∗tpw -sets of C5. Note that fγ∗tpw (C5) ≥ 2. It follows that C. Armada / Eur. J. Pure Appl. Math, 14 (2) (2021), 451-470 464 2 ≤ fγ∗tpw (C5) ≤ γ∗tpw (C5) = 2, that is, fγ∗tpw (C5) = 2. Now, suppose that n > 5. Let p = n 5 and j = 1, 2, . . . , p− 1, p. Group the vertices of Cn into p disjoint subsets Rj R1 = {u1, u2, u3, u4, u5} R2 = {u6, u7, u8, u9, u10} R3 = {u11, u12, u13, u14, u15} ... Rp−1 = {un−9, un−8, un−7, un−6, un−5} Rp = {un−4, un−3, un−2, un−1, un} Let i = 1, 6, 11, . . . , n − 4. For every induced subgraph 〈ui, ui+1, ui+2, ui+3, ui+4〉, the vertices ui, ui+1 form a total dr-power dominating set since ui+2 and ui+4 are directly observed vertices while ui+3 is a remotely observed vertex for all i = 1, 6, 11, . . . , n−9, n−4. Let the set R = {ui, ui+1 : i = 1, 6, 11, . . . , n− 9, n− 4} = {u1, u2, u6, u7, u11, u12, . . . , un−9, un−8, un−4, un−3} ,where |R| = 2p = 2(n 5 ) = 2n 5 , OR V (Cn) = V (Cn), OR E(Cn) = E(Cn), and the induced subgraph 〈R〉 has no isolated vertex. By Theorem 2.3, R is a γ∗tpw -set of Cn. Let m+ 1 be the number of γ∗tpw -sets of Cn where m is a positive integer. Let k = 1, 2, . . . ,m and Tk be a γ∗tpw -set of Cn different from R. Note that Tk can be formed by starting all the vertices of Sl for l = 1, 2, . . . , 5 and replacing u5 by un in S5. Now, if Tk starts with S1, then let k = 1 and {u1, u2} ⊆ T1. Replace u6 ∈ R by u5 to form T1. Then the next vertex to be choosen must be u6, that is, the vertex ui+4, ui must be in T1 for all i = 1, 6, 11, . . . , n − 9, n − 4. It follows that T1 = {u1, u2, u5, u6, u10, u11, u15, u16, . . . , un−5, un−4} such that |T1| = |R|. Since u1 ∈ T1 and un−4 is the last vertex in T1, by the same argument in Case 1, T1 is not a γ∗tpw -set of Cn. Since u6 is arbitrarily replaced from R, we cannot replace the vertex ui in R, where i = 6, 11, . . . , n− 9, n− 4 to form another γ∗tpw -set of Cn. Therefore, only the γ∗tpw -set R starts with S1 and so, {u1, u2} * Tk for all k = 1, 2, . . . ,m. Hence, {u1, u2} is a forcing subset for R. Therefore, fγ∗tpw (R) = 2 = fγ∗tpw (Cn). Case 3: Suppose that n ≡ 1(mod 5). By Theorem 2.3, γ∗tpw (Cn) = 2n+3 5 . Suppose that n = 6. Then γ∗tpw (C6) = 2(6)+3 5 = 3. Clearly, S1 = {u1, u2, u3}, S2 = {u2, u3, u4}, S3 = {u3, u4, u5}, S4 = {u4, u5, u6}, S5 = {u5, u6, u1}, S6 = {u6, u1, u2}, are the only γ∗tpw -sets of C6. Clearly, for l = 1, 2, . . . , 6, {u1, u3} ⊆ S1 and {u1, u3} * Sl for all l 6= 1. Thus, {u1, u3} is a forcing subset for S1, that is, fγ∗tpw (S1) = 2 = fγ∗tpw (C6). Now, suppose that n > 6. Let p = n−1 5 and j = 0, 1, 2, . . . , p− 1, p. Group the vertices of Cn into p+ 1 disjoint subsets Rj C. Armada / Eur. J. Pure Appl. Math, 14 (2) (2021), 451-470 465 R0 = {u1} R1 = {u2, u3, u4, u5, u6} R2 = {u7, u8, u9, u10, u11} R3 = {u12, u13, u14, u15, u16} ... Rp−1 = {un−9, un−8, un−7, un−6, un−5} Rp = {un−4, un−3, un−2, un−1, un} Let i = 2, 7, 12, . . . , n − 4. For every induced subgraph 〈ui, ui+1, ui+2, ui+3, ui+4〉, the vertices u1, ui, ui+1 form a total dr-power dominating set since ui+2 and ui+4 are directly observed vertices while ui+3 is a remotely observed vertex for all i = 2, 7, 12, . . . , n−9, n−4. Let the set R = {u1, ui, ui+1 : i = 2, 7, 12, . . . , n− 9, n− 4} = {u1, u2, u3, u7, u8, u12, u13, . . . , un−9, un−8, un−4, un−3} ,where |R| = 2p+ 1 = 2(n−1 5 ) + 1 = 2n+3 5 , OR V (Cn) = V (Cn), OR E(Cn) = E(Cn), and the induced subgraph 〈R〉 has no isolated vertex,that is, R is a γ∗tpw -set of Cn. Let m+ 1 be the number of γ∗tpw -sets of Cn where m is a positive integer. Let k = 1, 2, . . . ,m and Tk be a γ∗tpw -set of Cn different from R. Note that Tk can be formed by starting all the vertices of Sl for l = 1, 2, . . . , 5 and replacing u6 by un in S5 and S6 and also, Tk can be formed by having three vertices in any one or two of the Rj ’s where j ≥ 0 such that the induced subgraph is a graph P3 and two vertices in the other Rl’s where l 6= j. Replacing u3 ∈ R by u6 to form Tk, say T1, that is, T1 = {u1, u2, u6, u7, u8, u12, u13, . . . , un−9, un−8, un−4, un−3} is a γ∗tpw -set of Cn. Clearly, {u1, u3} * T1. Replacing u8 ∈ T1 by u11 to form Tk, say T2, that is, T2 = {u1, u2, u6, u7, u11, u12, u13, . . . , un−9, un−8, un−4, un−3} is a γ∗tpw -set of Cn. Clearly, {u1, u3} * T2. Continuing in this manner, {u1, u3} * Tk for some k. Now, if Tk starts with S1, then let k = 3 and {u1, u3} ⊆ T3. Then u2 must be in T3. Replace u7 ∈ R by u6 to form T3. Then the next vertex to be choosen must be u7, that is, the vertex ui+4, ui must be in T3 for all i = 2, 7, 12, . . . , n − 9, n − 4. Then T3 = {u1, u2, u3, u6, u7, u11, u12, u16, u17, . . . , un−5, un−4} such that |T3| = |R|. Since u1 ∈ T3 and un−4 is the last vertex in T3, by the same argument in Case 1, T3 is not a γ∗tpw -set of Cn. Since u7 is arbitrarily replaced from R, we cannot replace the vertex ui in R, where i = 7, 12, . . . , n− 9, n− 4 to form another γ∗tpw -set of Cn. Therefore, only the γ∗tpw -set R starts with S1 and so, {u1, u3} * Tk for all k = 1, 2, . . . ,m. Hence, {u1, u3} is a forcing subset for R. Therefore, fγ∗tpw (R) = 2 = fγ∗tpw (Cn). Case 4: Suppose that n ≡ 3(mod 5). By Theorem 2.3, γ∗tpw (Cn) = 2n+4 5 . Suppose that n = 8. Then γ∗tpw (C8) = 2(8)+4 5 = 4. Clearly, S1 = {u1, u2, u4, u5}, S2 = {u1, u2, u5, u6}, S3 = {u1, u2, u6, u7}, S4 = {u2, u3, u5, u6}, S5 = {u2, u3, u6, u7}, S6 = {u2, u3, u7, u8}, S7 = {u3, u4, u6, u7}, S8 = {u3, u4, u7, u8}, S9 = {u8, u1, u3, u4}, S10 = {u4, u5, u7, u8}, S11 = {u8, u1, u4, u5}, and S12 = {u8, u1, u5, u6}, are the only C. Armada / Eur. J. Pure Appl. Math, 14 (2) (2021), 451-470 466 γ∗tpw -sets of C8. Clearly, for l = 1, 2 . . . , 12, {u2, u4} ⊆ S1 and {u2, u4} * Sl for all l 6= 1. Thus, {u2, u4} is a forcing subset for S1, that is, fγ∗tpw (S1) = 2 = fγ∗tpw (C8). Now, suppose that n > 8. Let p = n−3 5 and j = 0, 1, 2, . . . , p− 1, p. Group the vertices of Cn into p+ 1 disjoint subsets Rj R0 = {u1, u2, u3} R1 = {u4, u5, u6, u7, u8} R2 = {u9, u10, u11, u12, u13} R3 = {u14, u15, u16, u17, u18} ... Rp−1 = {un−9, un−8, un−7, un−6, un−5} Rp = {un−4, un−3, un−2, un−1, un} Let i = 4, 9, 14 . . . , n−4. For every induced subgraph 〈ui, ui+1, ui+2, ui+3, ui+4〉, the vertices u1, u2, ui, ui+1 form a total dr-power dominating set since u3, ui+2 and ui+4 are directly observed vertices while ui+3 is a remotely observed vertex for all i = 4, 9, 14, . . . , n−9, n−4. Let the set R = {u1, u2, ui, ui+1 : i = 4, 9, 14, . . . , n− 9, n− 4} = {u1, u2, u4, u5, u9, u10, u14, u15, . . . , un−9, un−8, un−4, un−3} ,where |R| = 2p + 2 = 2(n−3 5 ) + 2 = 2n+4 5 , OR V (Cn) = V (Cn), OR E(Cn) = E(Cn), and the induced subgraph 〈R〉 has no isolated vertex. By Theorem 2.3, R is a γ∗tpw -set of Cn. Let m + 1 be the number of γ∗tpw -sets of Cn where m is a positive integer. Let k = 1, 2, . . . ,m and Tk be a γ∗tpw -set of Cn different from R. Note that Tk can be formed by starting all the vertices of Sl for l = 1, 2, . . . , 12 and replacing u8 by un in S9, S11 and S12. Now, if Tk starts with S1, then let k = 1 and {u2, u4} ⊆ {u1, u2, u4, u5} ⊆ T1. Replace u9 ∈ R by u8 to form T1. Then the next vertex to be choosen must be u9, that is, the vertex ui+4, ui must be in T1 for all i = 4, 9, 14, . . . , n − 9, n − 4. Then T1 = {u1, u2, u4, u5, u8, u9, u13, u14, u18, u19, . . . , un−5, un−4} such that |T1| = |R|. Since u1 ∈ T1 and un−4 is the last vertex in T1, by the same argument in Case 1, T1 is not a γ∗tpw -set of Cn. Since u9 is arbitrarily replaced from R, we cannot replace the vertex ui in R, where i = 9, 14, . . . , n− 9, n− 4 to form another γ∗tpw -set of Cn. Therefore, only the γ∗tpw -set R starts with S1 and so, {u2, u4} * Tk for all k = 1, 2, . . . ,m. Hence, {u2, u4} is a forcing subset for R. Therefore, fγ∗tpw (R) = 2 = fγ∗tpw (Cn). Case 5: Suppose that n ≡ 4(mod 5). By Theorem 2.3, γ∗tpw (Cn) = 2n+2 5 . Suppose that n = 9. Then γ∗tpw (C9) = 2(9)+2 5 = 4. Clearly, S1 = {u1, u2, u5, u6}, S2 = {u1, u2, u6, u7}, S3 = {u2, u3, u6, u7}, S4 = {u2, u3, u7, u8}, S5 = {u3, u4, u7, u8}, S6 = {u3, u4, u8, u9}, S7 = {u4, u5, u8, u9}, S8 = {u4, u5, u9, u1}, and S9 = {u5, u6, u9, u1} are the only γ∗tpw -sets of C9. Clearly, for l = 1, 2, . . . , 9, {u2, u5} ⊆ S1 and {u2, u5} * Sl for all l 6= 1. Thus, {u2, u5} is a forcing subset for S1, that is, fγ∗tpw (S1) = 2 = fγ∗tpw (C9). Now, suppose that n > 9. Let p = n−4 5 and j = 0, 1, 2, . . . , p− 1, p. Group the vertices of C. Armada / Eur. J. Pure Appl. Math, 14 (2) (2021), 451-470 467 Cn into p+ 1 disjoint subsets Rj R0 = {u1, u2, u3, u4} R1 = {u5, u6, u7, u8, u9} R2 = {u10, u11, u12, u13, u14} R3 = {u15, u16, u17, u18, u19} ... Rp−1 = {un−9, un−8, un−7, un−6, un−5} Rp = {un−4, un−3, un−2, un−1, un} Let i = 5, 10, 15, . . . , n − 4. For every induced subgraph 〈ui, ui+1, ui+2, ui+3, ui+4〉, the vertices u1, u2, ui, ui+1 form a total dr-power dominating set since u3,u4,ui+2 and ui+4 are directly observed vertices while ui+3 is a remotely observed vertex ∀ i = 5, 10, 15, . . . , n− 4. Let the set R = {u1, u2, ui, ui+1 : i = 5, 10, 15, . . . , n− 9, n− 4} = {u1, u2, u5, u6, u10, u11, u15, u16, . . . , un−9, un−8, un−4, un−3} ,where |R| = 2p+ 2 = 2(n−4 5 ) + 2 = 2n+2 5 , OR V (Cn) = V (Cn), OR E(Cn) = E(Cn), and the induced subgraph 〈R〉 has no isolated vertex. By Theorem 2.3, R is a γ∗tpw -set of Cn. Let m+ 1 be the number of γ∗tpw -sets of Cn where m is a positive integer. Let k = 1, 2, . . . ,m and Tk be a γ∗tpw -set of Cn different from R. Note that Tk can be formed by starting all the vertices of Sl for l = 1, 2, . . . , 9 and replacing u9 by un in S8 and S9. Now, if Tk starts with S1, then let k = 1 and {u2, u5} ⊆ {u1, u2, u5, u6} ⊆ T1. Replace u10 ∈ R by u9 to form T1. Then the next vertex to be choosen must be u10, that is, the vertex ui+4, ui must be in T1 for all i = 5, 10, 15, . . . , n−9, n−4. Then T1 = {u1, u2, u5, u6, u9, u10, u14, u15, . . . , un−5, un−4} such that |T1| = |R|. Since u1 ∈ T1 and un−4 is the last vertex in T1, by the same argument in Case 1, T1 is not a γ∗tpw -set of Cn. Since u10 is arbitrarily replaced from R, we cannot replace the vertex ui in R, where i = 10, 15, . . . , n− 9, n− 4 to form another γ∗tpw -set of Cn. Therefore, only the γ∗tpw -set R starts with S1 and so, {u2, u5} * Tk for all k = 1, 2, . . . ,m. Hence, {u2, u5} is a forcing subset for R. Therefore, fγ∗tpw (R) = 2 = fγ∗tpw (Cn). Theorem 3.6. Let n be a positive integer with n ≥ 2. Then the total dr-power domination number of the complete graph Kn is given by γ∗tpw (Kn) = 2 and its forcing total dr-power domination number is given by fγ∗tpw (Kn) = { 0, n = 2 2, n > 2. Proof. Let V (Kn) = {u1, u2, u3, . . . , un}. Clearly, each pair of vertices ui, uj such that i 6= j in Kn forms a γ∗tpw -set of Kn, and so, γ∗tpw (Kn) = 2. If n = 2, then K2 has exactly one γ∗tpw -set which is V (K2). By Theorem 3.1 (i), fγ∗tpw (K2) = 0. Suppose that n > 2. Note that for all i = 1, 2, . . . n, ui is contained in γ∗tpw -sets {ui, uj} and {ui, uk} such that i 6= j 6= k 6= i and so, the set {ui} is not a forcing subset for any γ∗tpw -set of Kn, that is, C. Armada / Eur. J. Pure Appl. Math, 14 (2) (2021), 451-470 468 fγ∗tpw (Kn) ≥ 2. Consequently, by Corollary 3.2, 2 ≤ fγ∗tpw (Kn) ≤ γ∗tpw (Kn) = 2. Therefore, fγ∗tpw (Kn) = 2 for all n > 2. Theorem 3.7. Let n be a positive integer with n ≥ 2. Then the total dr-power domination number of the fan graph Fn = K1 + Pn of order n + 1 is given by γ∗tpw (Fn) = 2 and its forcing total dr-power domination number is given by fγ∗tpw (Fn) = { 2, n = 2, 3, 1, n ≥ 4. Proof. By Corollary 2.6, γ∗tpw (Fn) = γ∗tpw (K1 + Pn) = 2. Let V (Fn) = {v, u1, u2, u3, . . . , un} such that deg(v) = n. Note that the γ∗tpw -sets of Fn are of the form {v, ui} for all ui ∈ V (Fn) and of the form {ui, uj} such that i 6= j and {ui, uj} is a γt-set of Pn by Theorem 2.5 and Corollary 2.6. Consider the following cases: Case 1: Suppose that either n = 2 or n = 3. By Proposition 2.4, γt(Pn) = 2 for n = 2, 3. If n = 2, then R1 = {v, u1}, R2 = {v, u2} and R3 = {u1, u2} are the γ∗tpw -sets of F2. If n = 3, then R1 = {v, u1}, R2 = {v, u2}, R3 = {v, u3}, R4 = {u1, u2} and R5 = {u2, u3} are the γ∗tpw -sets of F3. Clearly, for all ui ∈ V (Fn) and n = 2, 3, the singleton {ui}, together with {v}, is not contained in exactly one γ∗tpw -set of Fn, that is, the sets {ui} and {v} are not forcing subsets for any γ∗tpw -set of Fn. Thus, fγ∗tpw (Fn) ≥ 2. Then 2 ≤ fγ∗tpw (Fn) ≤ γ∗tpw (Fn) = 2. Therefore, fγ∗tpw (Fn) = 2 for n = 2, 3. Case 2: Suppose that n ≥ 4. By Proposition 2.4, γt(P4) = 2. If n = 4, then R1 = {v, u1}, R2 = {v, u2}, R3 = {v, u3}, R4 = {v, u4}, and R5 = {u2, u3} are the γ∗tpw -sets of F4. Clearly, {u1} ⊆ R1 and {u1} * Rl for l = 2, 3, 4, 5, that is, {u1} is a forcing subset for R1 and so, fγ∗tpw (R1) = 1 = fγ∗tpw (F4). If n > 4, then γt(Pn) > 2 by Proposition 2.4 and so, the γ∗tpw -sets of Fn are of the form {v, ui} for all ui ∈ V (Fn). Clearly, R = {v, u1} is the only γ∗tpw -set of Fn containing u1. Thus, {u1} is a forcing subset for R, that is, fγ∗tpw (R) = 1 = fγ∗tpw (Fn) for n > 4. Theorem 3.8. Let n be a positive integer with n ≥ 3. Then the total dr-power domination number of the wheel graph Wn = K1 + Cn of order n+ 1 is given by γ∗tpw (Wn) = 2 and its forcing total dr-power domination number is given by fγ∗tpw (Wn) = { 2, n = 3, 4 1, n ≥ 5. Proof. By Corollary 2.6, γ∗tpw (Wn) = γ∗tpw (K1 + Cn) = 2. Let V (Wn) = {v, u1, u2, u3, . . . , un} such that deg(v) = n. Note that the γ∗tpw -sets of Wn are of the form {v, ui} for all ui ∈ V (Wn) and of the form {ui, uj} such that i 6= j and {ui, uj} is a γt-set of Cn by Theorem 2.5 and Corollary 2.6. Consider the following cases: REFERENCES 469 Case 1: Suppose that either n = 3 or n = 4. By Proposition 2.4, γt(Cn) = 2 for n = 3, 4. If n = 3, then R1 = {v, u1}, R2 = {v, u2}, R3 = {v, u3}, R4 = {u1, u2}, R5 = {u2, u3} and R6 = {u1, u3} are the γ∗tpw -sets of W3. If n = 4, then R1 = {v, u1}, R2 = {v, u2}, R3 = {v, u3}, R4 = {v, u4}, R5 = {u1, u2}, R6 = {u2, u3}, R7 = {u3, u4} and R8 = {u4, u1} are the γ∗tpw -sets of W4. Clearly, for all ui ∈ V (Wn) and for n = 3, 4, the singleton {ui}, together with {v}, is not contained in exactly one γ∗tpw -set of Wn, that is, the sets {ui} and {v} are not forcing subsets for any γ∗tpw -set of Wn. Thus, fγ∗tpw (Wn) ≥ 2. Then 2 ≤ fγ∗tpw (Wn) ≤ γ∗tpw (Wn) = 2. Therefore, fγ∗tpw (Wn) = 2 for n = 3, 4. Case 2: Suppose that n ≥ 5. Then γt(Cn) > 2 by Proposition 2.4 and so, the γ∗tpw -sets of Wn are of the form {v, ui} for all ui ∈ V (Wn). Clearly, R = {v, u1} is the only γ∗tpw -set of Wn containing u1. Thus, {u1} is a forcing subset for R, that is, for all n ≥ 5, fγ∗tpw (R) = 1 = fγ∗tpw (Wn). Theorem 3.9. Let n be a positive integer with n ≥ 1. Then the total dr-power domination number of the star graph Sn = K1 +Kn of order n+ 1 is given by γ∗tpw (Sn) = 2 and its forcing total dr-power domination number is given by fγ∗tpw (Sn) = { 0, n = 1 1, n > 1. Proof. By Corollary 2.6, γ∗tpw (Sn) = γ∗tpw (K1 + Kn) = 2. Let V (Sn) = {v, u1, u2, u3, . . . , un} such that deg(v) = n. If n = 1, then R = {v, u1} is the only γ∗tpw -set of S1. By Theorem 3.1(i), fγ∗tpw (S1) = 0. If n > 1, then the γ∗tpw -sets of Sn are of the form {v, ui} for all ui ∈ V (Sn) by Theorem 2.5. Clearly, R = {v, u1} is the only γ∗tpw -set of Sn containing u1. Thus, {u1} is a forcing subset for R, that is, fγ∗tpw (R) = 1 = fγ∗tpw (Sn) for all n > 1. Acknowledgements The author thanks the peer reviewers of the paper and readers of European Journal of Pure and Applied Mathematics, for making the journal successful and to the Cebu Normal University for the financial support. The author also expresses warm gratitude to Ho JC for the emotional and moral support. References [1] D Amos. On total domination in graphs. University of Houston-Downtown, 2012. [2] C Armada. Forcing total dr-power domination number of graphs under some binary operations. European Journal of Pure and Applied Mathematics, accepted for publication. REFERENCES 470 [3] C Armada and S Canoy Jr. A-differential of graphs. 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