1-movable Restrained Domination in Graphs

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1 Global Journal of Pure and Applied Mathematics. ISSN Volume 12, Number 6 (2016), pp Research India Publications 1-movable Restrained Domination in Graphs Renario G. Hinampas, Jr., Jocecar Lomarda-Hinampas and Joann P. Reformina Colleges of Teacher Education and Advanced Studies, Bohol Island State University-Main Campus, CPG North Avenue, 6300 Tagbilaran City, Bohol, Philippines. Abstract Let G be a connected nontrivial graph. A nonempty subset S of V (G) is a 1- movable restrained dominating set of G if S is a restrained dominating set of G and for every v S, S \{v} is a restrained dominating set of G or there exists u (V (G) \ S) N G (v) such that (S \{v}) {u} is a restrained dominating set of G. The 1-movable restrained domination number of a graph G, denoted by γmr 1 (G), is the cardinality of the smallest 1-movable restrained dominating set of G. A 1-movable restrained dominating set of G with cardinality equal to γmr 1 (G) is called γmr 1 -set of G. This paper presents some properties of 1-movable restrained dominating set and investigates the 1-movable restrained dominating sets in the join of two graphs. Moreover, the bounds or exact values of the 1-movable restrained domination number are determined. AMS subject classification: 05C69. Keywords: Domination, restrained domination, 1-movable domination, 1-movable restrained domination. 1. Introduction Let G = (V (G), E(G)) be a graph and v V (G). The open neighborhood of v is the set N G (v) = N(v) ={u V (G) : uv E(G)} and the closed neighborhood of v is the set N G [v] =N[v] =N(v) {v}.

2 5246 Renario G. Hinampas, et al. A subset S of V (G) is a dominating set of G if for every v V (G)\S, there exists u S such that uv E(G). The domination number of G is denoted by γ (G) which refers to the cardinality of the smallest dominating set of G. A dominating set of G with cardinality equal to γ (G) is called a γ -set of G. A dominating set S is called a restrained dominating set of G if for every vertex u V (G) \ S has at least one neighbor in V (G) \ S. The restrained domination number of G is the cardinality of the smallest restrained dominating set of G and is denoted by γ r (G). A restrained dominating set with cardinality equal to γ r (G) is called γ r -set of G. A nonempty subset S of V (G) is a 1-movable dominating set of G if S is a dominating set of G and for every v S, S \{v} is a dominating set of G or there exists a vertex u (V (G)\S) N(v)such that (S \{v}) {u} is a dominating set of G. The 1-movable domination number of G denoted by γm 1 (G) is the cardinality of the smallest 1-movable dominating set of G. A 1-movable dominating set with cardinality equal to γm 1 (G) is called γm 1 -set of G. Considering only the movement of all the vertices in S, we call S as strictly 1-movable dominating set of G. A nonempty subset S of V (G) is a strictly 1-movable dominating set of G if S is a dominating set of G and for every v S, there exists a vertex u (V (G)\S) N(v)such that (S \{v}) {u} is a dominating set of G. The strictly 1-movable domination number of G denoted by γm 1 (G) is the cardinality of the smallest strictly 1-movable dominating set of G. A strictly 1-movable dominating set with cardinality equal to γm 1 (G) is called -set of G. γ 1 m A nonempty subset S of V (G) is a 1-movable restrained dominating set of G if S is a restrained dominating set of G and for every v S, S \{v} is a restrained dominating set of G or there exists u (V (G) \ S) N G (v) such that (S \{v}) {u} is a restrained dominating set of G. The 1-movable restrained domination number of a graph G, denoted by γmr 1, is the cardinality of the smallest 1-movable restrained dominating set of G. A 1-movable restrained dominating set of G with cardinality equal to γmr 1 (G) is called γmr 1 -set of G. Restrained dominating sets and their properties were investigated in [2], [3] and [5] while the 1-movable dominating sets and their corresponding properties were studied and developed in [1] and [6]. 2. Results The 1-movable restrained dominating set does not always exist in G. Define R 1 mr to be a collection of all graphs with 1-movable restrained dominating sets. This study focused on the graphs that belong to the collection R 1 mr. Theorem 2.1. [4] Let G and H be a connected graph. Then S V(G+ H) is a dominating set of G + H if and only if at least one of the following is true: (i) S V (G) is a dominating set of G. (ii) S V(H)is a dominating set of H.

3 1-movable Restrained Domination in Graphs 5247 (iii) S V (G) = φ and S V(H) = φ. Remark 2.2. For any connected nontrivial graph G, γ r (G) γ 1 mr (G). Remark 2.3. Let G be a graph of order n 3. If S is a 1-movable restrained dominating set of G, then 1 S n 2. Theorem 2.4. Let G be a connected nontrivial graph of order n 3. Then γ 1 mr (G) = 1 if and only if G = K 2 + H for some graph H. Proof. Assume that γmr 1 (G) = 1. Then G has a γ mr 1 -set say S = {a} for some a V (G). Since S is a 1-movable restrained dominating set there exists b (V (G) \ S) N G (a) such that (S \{a}) {b} ={b} is a restrained dominating set of G. Take V(K 2 ) = {a,b} and H = V (G) \ V(K 2 ). Then G = K 2 + H. For the converse, suppose G = K 2 + H. Let V(K 2 ) ={a,b} for some a,b V (G) and set S ={a}. Then S is a dominating set of G and V (G) \ S = {b} + H has no isolated vertex. Hence, S is a restrained dominating set of G. Moreover, there exists b N G (a) such that (S \{a}) {b} ={b} is a dominating set of G and V (G) \ {b} = {a} +H has no isolated vertex. Thus, (S\{a}) {b} = {b} is a restrained dominating set of G. This concludes that S is a 1-movable restrained dominating set of G. Consequently, (G) = S =1. γ 1 mr Corollary 2.5. For every complete graph K n of order n 3, γ 1 mr (K n) = 1. Theorem 2.6. Let G and H be connected nontrivial graphs of order m 2 and n 2, respectively. A nonempty subset of S of V(G+ H)with S m + n 2 is a 1-movable restrained dominating set of G + H if and only if one of the following holds: (i) S is a dominating set of G such that if S =1, then either S is a strictly 1-movable dominating set of G or {z} is a dominating set of H for some z V(H); (ii) S is a dominating set of H such that if S =1, then either S is a strictly 1-movable dominating set of H or {a} is a dominating set of G for some a V (G); or (iii) S = S 1 S 2 where S 1 V (G) and S 2 V(H)such that if (a) S 1 = V (G), then V(H)\ S 2 has no isolated vertex and for every v S 2, V(H)\ (S 2 \{v}) or V(H)\ (S 2 \{v}) {u} has no isolated vertex for some u (V (H ) \ S 2 ) N H (v) and if (b) S 2 = V(H), then V (G) \ S 1 has no isolated vertex and for every v S 1, V (G) \ (S 1 \{v}) or V (G) \ [(S 1 \{v}) {w}] has no isolated vertex for some w (V (G) \ S 1 ) N G (v). Proof. Assume that S is a 1-movable restrained dominating set of G + H. Suppose S V (G). Since S is a dominating set of G + H, S is a dominating set of G by Theorem 2.1(i). Suppose S = 1, say, S = {a} for some z V (G). Since S is

4 5248 Renario G. Hinampas, et al. a 1-movable restrained dominating set, there exists z [V(G+ H)\ S] N G+H (a) such that (S \ {a}) {z} = {z} is a dominating set of G + H. If z V (G), then (S \ {a}) {z} = {z} is a dominating set of G. This means that S is a strictly 1-movable dominating set of G. Ifz V(H), then {z} is a dominating set of H. Thus (i) holds. Similarly, (ii) holds if S V(H). Now, consider when S = S 1 S 2 where S 1 V (G) and S 2 V(H)and assume that S 1 = V (G). Since S is a restrained dominating set of G + H, V(G+ H)\ S = V(H)\ S 2 has no isolated vertex. Let v S 2. Suppose first that S \{v} is a restrained dominating set of G+H. Then V(G+ H)\ (S \{v}) = V(H)\ (S \{v}) has no isolated vertex. Suppose S \{v} is not a restrained dominating set of G + H. Then there exists u (V (H ) \ S) N H (v) such that (S \{v}) {u} is a restrained dominating set of G + H. Hence, V(G+ H)\ [(S \{v}) {u}] = V(H)\[(S 2 \{v}) {u}] has no isolated vertex. Similarly (iiib) holds if S 2 = V(H). For the converse, suppose first that (i) holds. By Theorem 2.1 (i), S is a dominating set of G + H. Clearly, V(G+ H)\ S has no isolated vertex. Hence, S is a restrained dominating set of G + H. Let v S. If S 2, then there exists u V(H)such that (S \ {v}) {u} is a dominating set of G + H and V(G+ H)\ [(S \ {v}) {u}] has no isolated vertex. Hence, (S \ {v}) {u} is a restrained dominating set of G + H.If S =1,then S = {a} for some a V (G). Suppose first that S is a strictly 1-movable dominating set of G. Then there exists w (V (G)\S) N G (a) such that (S\{a}) {w} = {w} is a dominating set of G. By Theorem 2.1(i), (S \ {a}) {w} = {w} is a dominating set of G + H. Moreover, it is clear to see that V(G+ H)\[(S \ {a})] {w} has no isolated vertex. Hence, (S \ {a}) {w} is a restrained dominating set of G + H. If S is not a strictly 1-movable dominating set of G. Then by assumption, {z} is a dominating set of H for some z V(H). Hence, (S \ {a}) {z} = {z} is a dominating set of G + H by Theorem 2.1(ii). Furthermore, V(G+ H)\ [(S \ {a}) {z}] has no isolated vertex. Thus, (S \ {a}) {z} is a restrained dominating set of G + H. Therefore, S is a 1-movable restrained dominating set of G + H. Similarly, if (ii) holds, then S is a 1-movable restrained dominating set of G + H. Suppose that (iii) holds. Then S is a dominating sets of G + H.If S 1 = V (G) and S 2 = V (G), then V(G+ H)\ S has no isolated vertex. Thus, S is a restrained dominating set of G + H. Let v S and suppose v S 1. If S 1 2, then S 1 \ {v} = φ. Thus, S \ {v} = (S 1 \ {v}) S 2 is a dominating set of G + H and V(G+ H)\[S \{v}] has no isolated vertex. Thus, S \{v} is a restrained dominating set of G+H. Suppose S 1 =1. Since G is connected, there exists u V (G) N G (v) such that S v = (S\{v}) {u} is a dominating set of G+H and V(G+ H)\ S v has no isolated vertex. Thus, S v is a restrained dominating set of G + H. Suppose that S 1 = V (G). Then S is a dominating set of G + H. By assumption S 2 n 2 and V(G+ H)\ S = V(H)\ S 2 has no isolated vertex. Hence, S is a restrained dominating set of G+H. Let v S. Suppose that v S 1. Since S 1 = V (G), S 1 \ {v} = φ. Thus, S \{v} =(S 1 \{v}) S 2 is a restrained dominating set of G + H. Suppose v S 2. Suppose first that V(G+ H)\ (S \{v}) = V(H)\ (S 2 \{v}) has no isolated vertex. Then, S \{v} =S 1 (S 2 \{v}) is a dominating set of G + H and hence a restrained dominating set of G + H. Suppose V(H)\ (S 2 \{v}) has isolated vertex. By assumption, V(H)\ [(S 2 \{v}) {u}] has no isolated vertex. Moreover,

5 1-movable Restrained Domination in Graphs 5249 the set (S \{v}) {u} =S 1 [(S 2 \{v}) {u}] for some u (V (G) \ S 2 ) N H (v), is a dominating set of G + H. Hence, (S \{v}) {u} is a restrained dominating set of G + H. Similar arguments follow when S 2 = V(H). Therefore, S is 1-movable restrained dominating set of G + H. Corollary 2.7. Let G and H be connected nontrivial graphs. Then { γmr 1 (G + H) = 1, if γ(g) = 1 = γ(h) or γm 1 1 (G) = 1orγm (H ) = 1 2, otherwise. Proof. Consider the following cases: Case 1: γ (G) = 1 and γ(h)= 1. Let S ={z} be a γ -set of G for some z V (G). Since γ(h)= 1, there exists w V(H) that dominates H. By Theorem 2.6 (i), S is a 1-movable restrained dominating set of G + H. Thus, γmr 1 (G + H) = S =1. Also, by Theorem 2.6 (ii), the set {w} is a γmr 1 -set of G + H. Thus, γ mr 1 (G + H) = {w} = 1. Case 2: γm 1 1 (G) = 1orγm (H ) = 1. Suppose first that γm 1 1 (G) = 1. Then G has a γm -set say, S ={z} for some z V (G). By Theorem 2.6 (i), S is a 1-movable restrained dominating set of G + H. Thus, γmr 1 1 (G + H) = S =1. Similarly if S ={d} is a γm -set of H for some d V(H), then S is a 1-movable restrained dominating set of G + H by Theorem 2.6 (ii) and so γmr 1 (G + H) = S =1. Case 3: γ (G) = 1 and γ(h) = 1. The graph G K 2 +H since γ (G) = 1 and γ(h) = 1. By Theorem 2.4, γmr 1 (G+H) 2. Let S ={x,y} be a γ -set of G. By Theorem 2.6 (i), S is a 1-movable restrained dominating set of G + H. Hence, 2 γmr 1 (G + H) S =2. Thus, γ mr 1 (G + H) = 2. Similarly, if S ={a,b} is a γ -set of H. Then by Theorem 2.6 (ii), S is a 1-movable restrained dominating set of G + H and γmr 1 (G + H) = S =2. Suppose S ={c, w} where c V (G) and w V(H). By Theorem 2.6 (iii), S is a 1-movable restrained dominating set of G + H. Thus, γmr 1 (G + H) = S =2. Theorem 2.8. Let H be a connected nontrivial graph of order n 2. A subset S of V(K 1 + H)with 1 S n 1 is a 1-movable restrained dominating set in K 1 + H if and only if one of the following holds: (i) S = V(K 1 ) and there exists z V(H)that dominates H ; (ii) S is a dominating set in H such that for every v S, S \ {v} or (S \ {v}) {u} is a dominating set of H or V(H)\ (S \ {v}) has no isolated vertex; or (iii) S = V(K 1 ) S 1, where 1 S 1 n 2 and

6 5250 Renario G. Hinampas, et al. (1) V(H)\ S 1 has no isolated vertex; (2) S 1 is a dominating set in H or S 1 {a} is a dominating set in H for some a V(H); and (3) for every v S 1, V(H)\ (S 1 \ {v}) or V(H)\[(S 1 \{v}) {u}] has no isolated vertex for some u (V (H ) \ S 1 ) N H (v). Proof. Assume that S is a 1-movable restrained dominating set of K 1 + H. Suppose that S = V(K 1 ) ={x}. Since S is a 1-movable restrained dominating set of K 1 + H, there exists z V(H)such that (S \{x}) {z} ={z} is a dominating set of K 1 + H. Hence, z dominates H. Thus, (i) holds. Suppose that S V(H). Let v S. Suppose first that S \{v} is a restrained dominating set of K 1 + H. Then S \{v} is a dominating set of K 1 + H and hence of H. Suppose S \{v} is not a restrained dominating set of K 1 + H. Then for some u [V(K 1 + H)\ S] N(v), (S \{v}) {u} is a restrained dominating set of K 1 + H.Ifu = x, then V(K 1 + H)\[(S \{v}) {u}] = V(H)\ (S \{v}) has no isolated vertex. If u = x, then u V(H)\ S. This means that, (S \{v}) {u} is a dominating set of K 1 + H and hence of H. Thus, (ii) holds. Suppose S = V(K 1 ) S 1. Since S is a restrained dominating set of K 1 + H, V(K 1 + H)\ S = V(H)\ S 1 has no isolated vertex. Let v S and assume that v = x. Suppose first that S \{x} =S 1 is a restrained dominating set of K 1 + H. Then S 1 is a dominating set of H. Suppose S \{x} is not a restrained dominating set of K 1 + H. Then there exists a V(H)\ S 1 such that (S\{x}) {a} =S 1 {a} is a restrained dominating set of K 1 +H and hence a dominating set of H. Suppose v = x. Then v S 1.IfS\{v} is a restrained dominating set of K 1 +H, then V(K 1 + H)\ (S \{v}) = V(H)\[S 1 \{v}] has no isolated vertex. Suppose that S \{v} is not a restrained dominating set. Then (S \{v}) {u} is a restrained dominating set of K 1 + H. Thus V(K 1 + H)\[(S \{v}) {u}] = V(H)\[(S 1 \{v}) {u}] has no isolated vertex. Hence (iii) holds. For the converse, suppose that (i) holds. Then S = V(K 1 ) = {x} is a dominating set of K 1 + H. Since H is a connected graph, V(K 1 + H)\ V(K 1 ) = H has no isloated vertex. Thus, S is a restrained dominating set of K 1 + H. By assumption, there exists z V(H) that dominates H. Hence, (S \{x}) {z} ={z} is a dominating set of H and hence of K 1 + H. Moreover, V(K 1 + H) \{z} = K 1 + V(H)\{z} has no isolated vertex. Hence, S is a 1-movable restrained dominating set of K 1 + H. Suppose that (ii) holds. Then S is a dominating set of K 1 + H and V(K 1 + H)\ S = K 1 + V(H)\ S 1 has no isolated vertex. Thus, S is a restrained dominating set of K 1 +H. Let v S. Suppose that S\{v} is a dominating set of H. Then V(K 1 + H)\ (S \{v}) = K 1 + V(H)\ (S \{v}) has no isolated vertex. Thus, S\{v} is a restrained dominating set of K 1 + H. Suppose S \{v} is not a dominating set of H. By assumption (S \{v}) {u} is a dominating set of H and hence of K 1 + H for some u (V (H ) \ S) N(v). Moreover, V(K 1 + H) \[(S \{v}) {u}] = K 1 + V(H)\ [(S \{v}) {u}] has no isolated vertex. Thus, (S \{v}) {u} is a restrained dominating set of K 1 + H. Suppose that (S \{v}) {u} is not dominating set of K 1 + H. By assumption, V(K 1 + H)\[(S \{v}) {x}] = V(H)\ (S \{v}) has no isolated vertex. Moreover, (S \{v}) {x} is a dominating set of K 1 + H. Hence,

7 1-movable Restrained Domination in Graphs 5251 (S \{v}) {x} is a dominating set of K 1 + H. Consequently, S is a 1-movable restrained dominating set of K 1 + H. Suppose that (iii) holds. By assumption S is a dominating set of K 1 + H,1 S 1 n 2and V(K 1 + H)\ S = V(H)\ S 1 has no isolated vertex. Thus, S is a restrained dominating set of K 1 + H. Let v S. Suppose that v = x. By assumption, S \{x} = S 1 is a dominating set of H and hence of K 1 + H and V(K 1 + H)\ (S \{x}) = {x} + V(H)\ S 1 has no isolated vertex. Thus, S \{x} is a restrained dominating set of K 1 + H. Suppose S 1 is not a dominating set of H. By assumption, S 1 {a} is a dominating set of H for some a (V (H ) \ S 1 ). Hence, (S \{x}) {a} =S 1 {a} is a dominating set of K 1 + H. Moreover, V(K 1 + H)\ [(S \{x}) {a}] = {x} + V(H)\ (S 1 {a}) has no isolated vertex. Therefore, (S \{x}) {a} is a restrained dominating set of K 1 +H. Suppose that v = x. Then v S 1. By assumption it directly follows that S \{v} or (S \{v}) {u} is a restrained dominating set of K 1 + H. Consequently, S is a 1-movable restrained dominating set of K 1 + H. Corollary 2.9. Let H be a connected nontrivial graph. Then (i) γ 1 mr (K 1 + H) = 1 if and only if γ(h)= 1; and (ii) γmr 1 (K 1 + H) γm 1 (H ). Remark The strict inequality in Corollary 2.9 (ii) can be attained, however, the given upper bound is sharp. Theorem Let m 3 and n 3 be integers. A subset S of V(K m,n ) with 1 S m + n 2 is a 1-movable restrained dominating set of K m,n if and only if S = S 1 S 2 with 2 S 1 m 1and 2 S 2 n 1. Proof. Suppose that S is a 1-movable restrained dominating set of K m,n with 1 S m+n 2. Let S 1 = S V( K m ) and S 2 = S V( K n ). Then S = S 1 S 2. Since S isa1- movable restrained dominating set of K m,n, S 1 = φ and S 2 = φ. Suppose that S 1 =1, say S 1 = {z} for some z V( K m ). Then, S 1 \{z} =S 2 is a restrained dominating set of K m,n. Since V( K n ) is an empty graph, S 2 = V( K n ). It is a contradiction since V( K m,n ) \ S 2 = K m an empty graph. Suppose S 1 \{z} is not a restrained dominating set. Then there exists a V( K n ) such that (S\{z}) {a} =(S 1 \{z}) (S 2 {a}) = S 2 {a} is a restrained dominating set of K m,n. Since V( K n ) is an empty graph, S 2 {a} = V( K n ). Hence, V(K m,n ) \ (S 2 {a}) is an empty graph which makes it a contradiction. Thus S 1 2. Suppose that S 1 =m. Then V(K m,n ) \ S = V( K n ) \ S 2 is a graph with all vertices are isolated. This contradicts the assumption. Therefore, 2 S 1 m 1. Similarly, 2 S 2 n 1. For the converse, suppose that S = S 1 S 2 with 2 S 1 m 1, 2 S 2 n 1 and 1 S m + n 2. Clearly, S is a restrained dominating set of K m,n. Let v S. Suppose that v S 1. Since, S 1 2, S 1 \{v} = φ and S \{v} =(S 1 \{v}) S 2 is clearly a restrained dominating set of K m,n. Similarly if v S 2, S \{v} =S 1 (S 2 \{v}) is a

8 5252 Renario G. Hinampas, et al. restrained dominating set of K m,n. Therefore, S is a 1-movable restrained dominating set of K m,n. Corollary Let m 3 and n 3 be integers. Then γ 1 mr (K m,n) = 4. References [1] J. Blair, R. Gera and S. Horton. Movable dominating sensor sets in networks. Journal of Combinatorial Mathematics and Combinatorial Computing, 77 (2011), [2] G.S. Domke, J.H. Hattingh, S.T. Hedetniemi, R.C. Laskar, and L.R. Markus, Restrained domination in graphs. Discrete Mathematics, 203(1999), [3] S.R. Canoy Jr., N. Tuan, and R.A. Namoco, Restrained independent dominating sets and some realization problems. International Journal of Mathematical Analysis, 8(2014), no. 42, [4] C.E. Go, and S. R. Canoy Jr., Domination in the corona and join of graphs. International Mathematical Forum, 6(2011), no. 16, [5] J.H. Hattingh, E. Jonck, E.J. Joubert, and A.R. Plummer, Nordhaus-Gaddum results for restrained domination and total restrained domination in graphs. (2006). [6] R.G. Hinampas,Jr., and S.R. Canoy, Jr. 1-movable domination in graphs. Applied Mathematical Sciences, 8 (2014), no. 172,