Int. J. Contemp. Math. Sciences, Vol. 3, 2008, no. 8, 383 - 387
Results on Total Restrained Domination in Graphs Nader Jafari Rad Department of Mathematics Shahrood University of Technology, Shahrood, Iran
[email protected] Abstract Let G = (V, E) be a graph. A set S ⊆ V (G) is a total restrained dominating set if every vertex of G is adjacent to a vertex in S and every vertex of V (G)\S is adjacent to a vertex in V (G)\S. The total restrained domination number of G, denoted by γtr (G), is the smallest cardinality of a total restrained dominating set of G. In this paper we continue the study of total restrained domination in graphs and obtain some new results.
Mathematics Subject Classification: 05C69 Keywords: Domination; total domination; restrained domination
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Introduction
A vertex in a graph G dominates itself and its neighbors. A set of vertices S in a graph G is a dominating set, if each vertex of G\S is dominated by some vertex of S. The domination number γ(G) of G is the minimum cardinality of a dominating set of G. A dominating set S is called total dominating set if each vertex of G is dominated by some vertices of S, and the total domination number of G denoted by γt (G) is the minimum cardinality of a total dominating set of G,[3]. An end-vertex in a graph G is a vertex of degree one and a support vertex is one that is adjacent to an end-vertex. A dominating set S in a graph G is called a restrained dominating set in G, if each vertex x ∈ V (G)\S is adjacent both to a vertex y ∈ S and to a vertex z ∈ V (G)\S. The restrained domination number of G denoted by γr (G) is the minimum cardinality of a restrained dominating set of G. Johannes H. Hattingh et al in [2] studied total restrained domination in trees in [2]. A set S ⊆ V (G) is a total restrained dominating set if every
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vertex is adjacent to a vertex in S and every vertex of V (G)\S is adjacent to a vertex in V (G)\S. The total restrained domination number of G, denoted by γtr (G), is the smallest cardinality of a total restrained dominating set of G. A total dominating set of cardinality γt (G) we call a γt (G)-set and a total restrained dominating set of cardinality γtr (G) we call a γtr (G)-set. In this paper we continue the study of total restrained domination in graphs. We determine the total restrained domination number in some families of graphs. We also verify some bounds for total restrained domination number. We then study the affection of total restrained domination number by adding a pendant vertex or subdividing an edge. In this paper we denote the cartesian product of two graphs G and H by G2H and the cross product of them by G × H and adopt the definitions of [3].
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Results
Since any total restrained dominating set is also a total dominating set, so the inequality γt (G) ≤ γtr (G) for any graph G is obvious. Now let G be a graph with γt (G) = m and let v ∈ V (G) belongs to a γt (G)-set. Let G be a graph obtained from G by adding k pendant vertex to v, then γt (G ) = m and γtr (G ) = m + k. Hence the difference γtr (G) − γt (G) for a graph G can be arbitrarily large. On the other hand if a graph G has a γt (G)-set S such that G[V (G)\S] has no-isolate vertices, then S is also a total restrained dominating set, so γtr (G) ≤ γt (G). Hence we have the following characterization of those graphs for which the total restrained domination number is equal to the total domination number. Theorem 1 For a graph G, γt (G) = γtr (G) if and only if G has a γt (G)-set S such that G[V (G)\S] has no-isolate vertices. Now we use Theorem 1 to determine the total restrained domination number in some families of graphs. 3 n=3 , Theorem 2 1) For any n ≥ 2, γtr (Kn ) = 2 n>3 2 min{m, n} = 1 2) For any m, n, γtr (Km,n ) = , m + n min{m, n} = 1 3) For any n1 , n2 , ..., nm , γtr (Kn1 ,n2 ,...,nm) = 2, min{n1 , n2 , ..., nm } = 1, , 4) For any n ≥ 4, γtr (Cn ) = n − 2 n−2 4 5) For any n ≥ 4, γtr (Wn ) = 2, 6) If G is a connected graph, then γtr (cor(G)) = 2 |V (G)| , and for Grid graphs
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7) For any n ≥ 3, γtr (K2 2Pn ) = 2 8) For any n ≥ 3, γtr (P3 2Pn ) = n,
n 3
,
6n+8 n ≡ 1, 2, 4(mod5) 9) For any n ≥ 4, γtr (P4 2Pn ) = 6n+85 , + 1 n ≡ 0, 3(mod5) 5 10) If k and n are two integers greater than 16 then, 3kn+2(k+n)) (k+2)(n+2) − 1 ≤ γtr (Pk 2Pn ) ≤ − 4, 12 4 11) In a graph G any end-vertex and any support vertex belong to any γtr (G)-set, 12) For any two graphs G and H, γtr (G × H) ≤ γtr (G)γtr (H). Proof. We only prove part 12. The proof of other parts follows from Theorem 1 and the results of [1,3]. Let D be a γtr (G)-set and D be a γtr (H)set. We show that D × D is a total restrained dominating set for G × H. Let (u, v) ∈ G × H. There exist a ∈ D, b ∈ D such that a is adjacent to u and b is adjacent to v. Now (a, b) ∈ D×D dominates (u, v). Let (u , v ) ∈ G×H\D×D , then there exist a ∈ D, b ∈ D such that a is adjacent to u and b is adjacent to v . So (a , b ) is adjacent to (u , v ). Theorem 3 If T is a tree, then γtr (T ) ≥ 1 + (T ). Moreover the equality holds if and only if T is a star. Proof. Let T be a tree and let S be a γtr (T )-set. By Theorem 2 (11), S contains any end-vertex and any support vertex of T . Since T has at least (T ) end-vertices, then |S| ≥ 1 + (T ). For the other statement if T is a star, then the result follows from Theorem 2 (2). So let T be a tree with γtr (T ) = 1 + (T ). We show that T is a star. Let S be a γtr (T )-set with size |S| = 1 + (T ) and let v be a vertex of T with deg(v) = (T ). If v ∈ / S, then |S| ≥ 2(T ) which is impossible, so v ∈ S. Now it follows from |S| = 1+(T ) that N(v) ⊆ S and any vertex of N(v) is an end-vertex. Hence S is a star. Note that the above bound is strict. In the following we determine an upper bound and a lower bound for the total restrained domination number in terms of diameter.
diam(G)+2 diam(G)−1 Proposition 4 For any graph G, ≤ γtr (G) ≤ n − 2 . 4 4 It folProof. Let Pd+1 be the longest path in G with length d = diam(G). d+1−2 diam(G)−1 . lows from [2] that γtr (Pd+1 ) = d + 1 − 2 , so γtr (G) ≤ n − 2 4 4 On the other hand it is easy to see that diam(G) ≤ 2γtr (G) − 1 when γtr (G) is even and diam(G) ≤ 2γtr (G) − 2 when γtr (G) is odd. Hence the result follows. Now we study how total restrained domination number is affected by adding a pendant vertex. Let G be a graph obtained from G by adding a pendant
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vertex u and joining u to v. If v belongs to some γtr (G)-sets then γtr (G ) ≤ γtr (G) + 1 and this bound is both sharp and strict. On the other hand if v doesn’t belong to a γtr (G)-set for any γtr (G)-set, then γtr (G ) can be larger than or smaller than γtr (G). In the following we show that the addition a pendant vertex to a graph G may arbitrarily increase or decrease the total restrained domination number. Theorem 5 For any positive integer k ≥ 1, there are two graphs G and G for which G is obtained from G by adding a pendant vertex and |γtr (G ) − γtr (G)| = k + 1. Proof. Let V (P4 ) = {v1 , v2 , v3 , v4 } and let k ≥ 1 be an integer. For each i = 1, 2, ..., k, we add an ear v1 wiv3 to P4 to obtain a graph G. Then γtr (G) = 3. Now let G be a graph obtained from G by adding a pendant vertex to v1 , then γtr (G ) = k + 4. Hence γtr (G ) − γtr (G) = k + 1. On the other hand let V (C6 ) = {w1 , w2, w3 , w4 , w5 , w6 }. For each i = 1, 2, ...k we add the ears w1 xi w3 and w1 yi w5 and also add a pendant edge w1 z to obtain a graph G. Then γtr (G) = k + 5. Now let G be a graph obtained from G by adding the pendant edge w4 t, then γtr (G ) = 4. Hence γtr (G) − γtr (G ) = k + 1 By a subdivision of an edge uv we mean removing edge uv, and adding a new vertex x, and adding two edges ux and vx. Let G be a graph obtained from G by subdividing an edge uv. If at least one of u or v belongs to a γtr (G)-set, then γtr (G ) ≤ γtr (G) + 1. But in the general case there is no upper bound for γtr (G ) in terms of γtr (G). Proposition 6 For any positive integer k ≥ 1 there are two graphs G and G such that G is obtained from G by subdividing an edge of G and γtr (G ) − γtr (G) = k + 1. Proof. Let V (P8 ) = {v1 , v2 , v3 , v4 , v5 , v6 , v7 , v8 } and let k ≥ 1 be an integer. For each i = 1, 2, ..., k, we add two ears v2 wi v4 , v5 yi v7 to P8 to obtain a graph G. Then it is easy to see that γtr (G) = 6. Now let G be a graph obtained from G by subdividing the edge v4 v5 . Then γtr (G ) = k+7. Hence γtr (G )−γtr (G) = k+1.
References [1] Sylvain Gravier, Total domination number of grid graphs, Discrete Applied Mathematics 121 (2002), 119-128. [2] Johannes H. Hattingh, Elizabeth Jonck, Ernst J. Joubert and Andrew R. Plummer, Total restrained domination in trees, Discrete Math. 307 (2007), 1643-1650.
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[3] T. W. Haynes, S. T. Hedetniemi, and P. J. Slater, editors. Fundamental of domination in graphs: Advanced Topics. (Marcel Dekker, Inc, New York, NY), 1998. Received: September 29, 2007
