EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 14, No. 3, 2021, 1098-1107 ISSN 1307-5543 – ejpam.com Published by New York Business Global Forcing Total dr-Power Domination Number of Graphs Under Some Binary Operations Cris L. Armada1,∗ 1 Mathematics Department, College of Arts and Sciences, Cebu Normal University, Cebu City, Philippines 6000 Abstract. In this paper, the total dr-power domination number of graphs such as complete bipartite graph, generalized fan and generalized wheel are obtained. The forcing total dr-power domination number of graphs resulting from some binary operations such as join, corona and lexicographic product of graphs were determined. 2020 Mathematics Subject Classifications: 05C69 Key Words and Phrases: Forcing, total, dr-power domination, join, corona, lexicographic product 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. Some measurement devices must be placed at selected locations so that all the state variables of the system can be measured in order to monitor the power system. 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. The graphs considered in this paper are simple, connected, 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. ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v14i3.3915 Email addresses: armadac@cnu.edu.ph / cris.armada@g.msuiit.edu.ph (C. Armada) http://www.ejpam.com 1098 c© 2021 EJPAM All rights reserved. C. Armada / Eur. J. Pure Appl. Math, 14 (3) (2021), 1098-1107 1099 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 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 . 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 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 corona G ◦H of two graphs G and H is the graph obtained by taking one copy of G and |V (G)| copies of H, and then forming the join 〈{v}〉+Hv = v +Hv, where Hv is a copy of H, for each v ∈ V (G). C. Armada / Eur. J. Pure Appl. Math, 14 (3) (2021), 1098-1107 1100 The lexicographic product (composition) G[H] of two graphs G and H is the graph with V (G[H]) = V (G)× V (H), and (u, u′)(v, v′) ∈ E(G[H]) if and only if either uv ∈ E(G) or u = v and u′v′ ∈ E(H). Amos [1] studied total domination. The relation between forcing and domination concepts was investigated by Chartrand et al. [5] and they defined "forcing domination number". 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] was studied by Canoy, et al. The total dr-power domination number of some special graphs such as paths and cycles was studied by Armada [2]. 2. Known Results This section contains known results involving total dr-power domination, dr-power domination, total domination numbers of a graph G that are very useful in proving the main results of this paper. Remark 2.1. [6] For a graph G without isolated vertices, γ∗pw(G) ≤ γ∗tpw (G) ≤ γt(G). Proposition 2.2. [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 2.3. [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 2.4. [6] Let G and H be any graphs. Then γ∗tpw (G+H) = 2. C. Armada / Eur. J. Pure Appl. Math, 14 (3) (2021), 1098-1107 1101 Theorem 2.5. [6] Let G be a nontrivial connected graph and H be a graph with no isolated vertex. Then P ⊆ V (G ◦H) is a total dr-power dominating set if and only if P = A ⋃(⋃ v∈A Bv )⋃⋃ u/∈A Du  where A ⊆ V (G), Bv ⊆ V (Hv) for each v ∈ A and Bv 6= ∅ for each v /∈ NG(A), and Du ⊆ V (Hu) is a total dominating set of Hu for each u /∈ A. Corollary 2.6. [6] Let G be a nontrivial connected graph of order m and H be any graph with no isolated vertex. Then γ∗tpw (G ◦H) = m. Theorem 2.7. [6] Let G and H be nontrivial connected graphs. Then P = ⋃ x∈S ({x} × Tx), where S ⊆ V (G) and Tx ⊆ V (H) for all x ∈ S, is a total dr- power dominating set of G[H] if and only if S is a dominating set of G, and Tx is a total dominating set of H for each x ∈ S\N(S). Corollary 2.8. [6] Let G and H be nontrivial connected graphs. Then P is a total dr-power dominating set of G[H] if and only if P is a total dominating set of G[H]. Moreover, γ∗tpw (G[H]) = γt(G). Theorem 2.9. [2] 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. Corollary 2.10. [2] Let G be a connected graph. Then 0 ≤ fγ∗tpw (G) ≤ γ∗tpw (G). Theorem 2.11. [2] 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. C. Armada / Eur. J. Pure Appl. Math, 14 (3) (2021), 1098-1107 1102 3. Forcing Total dr-Power Domination Number of the Join of Graphs This section contains the total dr-power domination number of the complete bipartite graphs, generalized fan graphs, generalized wheel graphs, Pn + Pm, Pn +Cm and Cn +Cm and their forcing total dr-power domination numbers. Corollary 3.1. Let G and H be any graphs. Then R ⊆ V (G+H) is a γ∗tpw -set of G+H if and only if at least one of the following holds: (i) R is a γt-set of G and |R| = 2, (ii) R is a γt-set of H and |R| = 2, (iii) |R ∩ V (G)| = 1 and |R ∩ V (H)| = 1. Theorem 3.2. For any graphs G and H, fγ∗tpw (G+H) =  0, if G and H are both trivial, 1, if G is trivial and (i) H has an isolated vertex or (ii) γt(H) > 2 or (iii) γt(H) = 2 and there exists a vertex in H which is not in any γt-set of H, or if H is trivial and (i) G has an isolated vertex or (ii) γt(G) > 2 or (iii) γt(G) = 2 and there exists a vertex in H which is not in any γt-set of G, 2, otherwise. Proof. Consider the following cases: Case 1: G and H are both trivial graphs. {x, y} such that x ∈ V (G) and y ∈ V (H) is the only γt-set of G + H by Corollary 3.1. Thus, fγ∗tpw (G+H) = 0 by Theorem 2.9(i). Case 2: G is trivial and (i) H has an isolated vertex or (ii) γt(H) > 2 or (iii) γt(H) = 2 and H contains a vertex which is not in any γt-set of H Let V (G) = {x}. Suppose that H has an isolated vertex, say w. By Corollary 3.1, Rw = {x,w} is the only γ∗tpw -set of G+H containing w. If γt(H) > 2, then by Corollary 3.1, for each u ∈ V (H), Ru = {x, u} is the only γ∗tpw -set of G+H containing u. If γt(H) = 2 and there exists a vertex, say v, in H which is not in any γt-set of H, then by Corollary 3.1, Rv = {x, v} is the only γ∗tpw -set of G + H containing v. Thus, in any of the three cases, there always exists a vertex which is contained in exactly one γ∗tpw -set of G + H. By Theorem 2.9(ii), fγ∗tpw (G+H) = 1. Similarly, if H is trivial and (i) G has an isolated vertex or (ii) γt(G) > 2 or (iii) γt(G) = 2 and there exists a vertex in G which is not in any γt-set of G, then fγ∗tpw (G+H) = 1. C. Armada / Eur. J. Pure Appl. Math, 14 (3) (2021), 1098-1107 1103 Case 3: G is trivial , γt(H) = 2 and every vertex v ∈ V (H) is contained in a γt-set of H Let V (G) = {x}. By Corollary 3.1, for each u ∈ V (H), Ru = {x, u} is a γ∗tpw -set of G+H. Also by assumption and Corollary 3.1, for all v, w ∈ V (H) such that v 6= w, R = {v, w} is a γt-set of H and a γ∗tpw -set of G + H. Clearly, no single element is contained in exactly one γ∗tpw -set of G+H, that is, fγ∗tpw (G+H) ≥ 2. Consequently, by Corollary 2.10, 2 ≤ fγ∗tpw (G+H) ≤ γ∗tpw (G+H) = 2. Therefore, fγ∗tpw (G+H) = 2. Similarly, ifH is trivial , γt(G) = 2 and every vertex v ∈ V (G) is contained in a γt-set of G, then fγ∗tpw (G+H) = 2. Case 4: G and H are both nontrivial graphs. By Corollary 3.1, for each x ∈ V (G) and for each u ∈ V (H), R = {x, u} is a γ∗tpw -set of G+H. Thus, for each u ∈ R, there exists uy ∈ V (G+H)\R such that [R\{u}] ∪ {uy} is a γ∗tpw -set of G+H. By Theorem 2.11, it follows that fγ∗tpw (G+H) = γ∗tpw (G+H) = 2. The next result is a direct consequence of Corollary 2.4 and Theorem 3.2. Corollary 3.3. For any graph H, the total dr-power domination number of the join K1+H is given by γ∗tpw (K1 +H) = 2 and its forcing total dr-power domination number is given by fγ∗tpw (K1 +H) =  0, if H is trivial, 1, if H has an isolated vertex, or γt(H) > 2, or γt(H) = 2 and there exists a vertex in H which is not in any γt-set of H, 2, otherwise. Corollary 3.4. The total dr-power domination number of the complete bipartite Kn,m = Kn + Km such that n,m ≥ 1, is given by γ∗tpw (Kn,m) = 2 and its forcing total dr-power domination number is given by fγ∗tpw (Kn,m) =  0, n = 1 and m = 1, 1, n = 1 and m ≥ 2 or m = 1 and n ≥ 2, 2, n ≥ 2 and m ≥ 2. Proof. By Corollary 2.4, γ∗tpw (Kn,m) = γ∗tpw (Kn +Km) = 2. If n = 1 and m = 1, then K1 = K1 is trivial, and so by Corollary 3.3, fγ∗tpw (K1,1) = 0. If n = 1 and m ≥ 2, then Km has an isolated vertex, and so by Corollary 3.3, fγ∗tpw (K1,m) = 1. Similarly, if m = 1 and n ≥ 2, then fγ∗tpw (Kn,1) = 1. If n ≥ 2 and m ≥ 2, then Kn and Km are nontrivial graphs and so, by Theorem 3.2, fγ∗tpw (Kn,m) = 2. C. Armada / Eur. J. Pure Appl. Math, 14 (3) (2021), 1098-1107 1104 Corollary 3.5. The total dr-power domination number of the generalized fan Fn,m = Kn + Pm, where n ≥ 1 and m ≥ 2, is given by γ∗tpw (Fn,m) = 2 and its forcing total dr-power domination number is given by fγ∗tpw (Fn,m) = { 1, n = 1 and m ≥ 4, 2, otherwise. Proof. By Corollary 2.4, γ∗tpw (Fn,m) = γ∗tpw (Kn + Pm) = 2. Let Pn = [u1, u2, . . . , un]. If n = 1 and m = 4, then by Proposition 2.2, γt(P4) = 2 and P4 has exactly one γt-set which is {u2, u3}, that is, u1 is a vertex not in a γt-set of P4. By Corollary 3.3, γ∗tpw (F1,4) = 1. If n = 1 and m > 4, then by Proposition 2.2, γt(Pm) > 2. By Corollary 3.3, γ∗tpw (F1,m) = 1. If n = 1 and m < 4, then by Proposition 2.2, γt(P2) = γt(P3) = 2 and so, {u1, u2} is the γt-set of P2 while {u1, u2} and {u2, u3} are γt-sets of P3. Clearly, for m = 2, 3, every vertex ui ∈ V (Pm) is contained in a γt-set of Pm. By Corollary 3.3, γ∗tpw (F1,m) = 2. If n ≥ 2 and m ≥ 2, then Kn and Pm are nontrivial graphs. By Corollary 3.3, fγ∗tpw (Fn,m) = 2. Corollary 3.6. The total dr-power domination number of the generalized wheel Wn,m = Kn + Cm, where n ≥ 1 and m ≥ 3, is given by γ∗tpw (Wn,m) = 2 and its forcing total dr-power domination number is given by fγ∗tpw (Wn,m) = { 1, n = 1 and m ≥ 5, 2, otherwise. Proof. By Corollary 2.4, γ∗tpw (Wn,m) = γ∗tpw (Kn + Cm) = 2. Let Cn = [un, u1, u2, . . . , un]. If n = 1 and m ≥ 5, then by Proposition 2.2, γt(Cm) > 2. By Corollary 3.3, γ∗tpw (W1,m) = 1. If n = 1 and m < 5, then by Proposition 2.2, γt(C3) = γt(C4) = 2 and so, {u1, u2}, {u2, u3} and {u3, u1} are the γt-sets of C3 while {u1, u2}, {u2, u3}, {u3, u4} and {u4, u1} are γt-sets of C4. Clearly, form = 3, 4, every vertex ui ∈ V (Cm) is contained in a γt-set of Cm. By Corollary 3.3, γ∗tpw (W1,m) = 2. If n ≥ 2 and m ≥ 3, then Kn and Cm are nontrivial graphs. By Corollary 3.3, fγ∗tpw (Wn,m) = 2. Corollary 3.7. The total dr-power domination number of the join Pn + Pm, where n ≥ 1 and m ≥ 1, is given by γ∗tpw (Pn + Pm) = 2 and its forcing total dr-power domination number is given by fγ∗tpw (Pn + Pm) =  0, n = 1 and m = 1, 1, either n = 1 and m ≥ 4, or m = 1 and n ≥ 4, 2, otherwise. Proof. By Corollary 2.4, γ∗tpw (Pn + Pm) = 2. If n = 1 and m = 1, then P1 = K1 is a trivial graph and so, by Corollary 3.3, γ∗tpw (P1 + P1) = 0. If n = 1 and m ≥ 4, then fγ∗tpw (P1 + Pm) = fγ∗tpw (F1,m) = 1 by Corollary 3.5. Similarly, if m = 1 and n ≥ 4, fγ∗tpw (Pn + P1) = 1. If n = 1 and either m = 2 or m = 3, then by Corollary 3.5, C. Armada / Eur. J. Pure Appl. Math, 14 (3) (2021), 1098-1107 1105 fγ∗tpw (P1 + Pm) = fγ∗tpw (F1,m) = 2. Similarly, if m = 1 and either n = 2 or n = 3, then fγ∗tpw (Pn + P1) = 2. If n ≥ 2 and m ≥ 2, then Pn and Pm are nontrivial graphs. By Corollary 3.3, fγ∗tpw (Pn + Pm) = 2. Corollary 3.8. The total dr-power domination number of the join Pn + Cm, where n ≥ 1 and m ≥ 3, is given by γ∗tpw (Pn + Cm) = 2 and its forcing total dr-power domination number is given by fγ∗tpw (Pn + Cm) = { 1, n = 1 and m ≥ 5 2, otherwise. Proof. By Corollary 2.4, γ∗tpw (Pn + Cm) = 2. If n = 1 and m ≥ 5, then by Corollary 3.6, fγ∗tpw (P1 + Cm) = fγ∗tpw (W1,m) = 1. If n = 1 and m < 5, then by Corollary 3.6, fγ∗tpw (P1 + Cm) = fγ∗tpw (W1,m) = 2. If n ≥ 2 and m ≥ 2, then Pn and Cm are nontrivial graphs. By Corollary 3.3, fγ∗tpw (Pn + Cm) = 2. Corollary 3.9. The total dr-power domination number of the join Cn + Cm, where n ≥ 3 and m ≥ 3, and its forcing total dr-power domination number is given by fγ∗tpw (Cn + Cm) = γ∗tpw (Cn + Cm) = 2. Proof. By Corollary 2.4, γ∗tpw (Cn + Cm) = 2. Note that the cycles Cn and Cm are nontrivial graphs. By Theorem 3.2, fγ∗tpw (Cn + Cm) = 2. 4. Forcing Total dr-Power Domination Number of the Corona of Graphs This section contains the forcing total dr-power domination number of the coronas G ◦H and Km ◦H such that G is a nontrivial connected graph, Km is a complete graph and H is any graph. Theorem 4.1. Let G be a nontrivial connected graph and let H be any graph. Then R ⊆ V (G ◦H) is a γ∗tpw -set of G ◦H if and only if R = V (G). In particular, fγ∗tpw (G ◦H) = 0. Proof. Suppose that R ⊆ V (G ◦ H) is a γ∗tpw -set of G ◦ H. Note that V (G) is a γ∗tpw -set of G◦H by Corollary 2.6. Suppose that R 6= V (G) and let A = R∩V (G), that is, |A| < |V (G)|. Since R is a γ∗tpw -set of G ◦H, R = A ∪ (⋃ v∈A Bv ) ∪ ⋃ u/∈A Du  as described in Theorem 2.5 where |Bv| = 0 for each v ∈ A and |Du| = γt(H) for each u /∈ A or u ∈ V (G)\A . Hence, C. Armada / Eur. J. Pure Appl. Math, 14 (3) (2021), 1098-1107 1106 |R| = |A|+ γt(H)(|V (G)| − |A|) ≥ |A|+ 2|V (G)| − 2|A| since γt(H) ≥ 2 ≥ 2|V (G)| − |A| > |V (G)| = m since |V (G)| > |A| This is a contradiction since |R| = m by Corollary 2.6. Thus, R = V (G). The converse is clear. In particular, since V (G) is the unique γ∗tpw -set of G ◦ H, by Theorem 2.9(i), fγ∗tpw (G ◦H) = 0. The next result follows directly from Corollary 3.3 and Theorem 4.1. Note that K1 ◦H = K1 +H. Corollary 4.2. Let Km be a complete graph of order m ≥ 1 and let H be any graph. Then fγ∗tpw (Km ◦H) =  0, if either m = 1 and H is trivial, or m > 1, 1, if m = 1 and either (i) H has an isolated vertex, or (ii) γt(H) > 2, or (iii) γt(H) = 2 and there exists a vertex in H which is not in any γt-set of H, 2, if m = 1, γt(H) = 2 and every vertex in H is contained in any γt-set of H. 5. Forcing Total dr-Power Domination Number of the Lexicographic Product of Graphs This section contains the forcing total dr-power domination number of the graphs G[H], Pn[H] and Cn[H] where G and H are nontrivial connected graphs. Theorem 5.1. Let G and H be nontrivial connected graphs. Then fγ∗tpw (G[H]) = γt(G). Proof. Note that γ∗tpw (G[H]) = γt(G) by Corollary 2.8. Now, suppose that P = ⋃ u∈S ({u} × Tu), where S is a γt-set of G and Tu ⊆ V (H). Consequently, |P | = |S| = γt(G). By Theorem 2.7 and Corollary 2.8, P is a γ∗tpw -set of G[H]. Sup- pose that fγ∗tpw (G[H]) = fγ∗tpw (P ). Moreover, suppose that P has a forcing subset R with |R| < |P |, that is, P = R ∪ N , where N = {(u, v) ∈ P : (u, v) /∈ R}. Pick (u, v) ∈ N . Note that there always exists a vertex (u,w) ∈ V (G[H])\P such that w 6= v and [P\{(u, v)}] ∪ {(u,w)} = Q is a γ∗tpw -set of G[H] since all adjacent vertices (r, s) REFERENCES 1107 of (u, v) with u 6= r are adjacent vertices of (u,w) also. Thus, Q = R ∪ M , where M = [N\{(u, v)}] ∪ {(u,w)}, OQ V (G[H]) = V (G[H]) and OQ E(G[H]) = E(G[H]) , 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[H]) = |P | = γt(G). The next result follows from Theorem 5.1 and Corollary 2.2. Corollary 5.2. Let H be a nontrivial connected graph and n ≥ 3. Then fγ∗tpw (Pn[H]) = fγ∗tpw (Cn[H]) =  n 2 , n ≡ 0(mod 4), n+2 2 , n ≡ 2(mod 4), n+1 2 , otherwise. 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 subsets for γ∗tpw -sets in graphs. 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