J. Comp. & Math. Sci. Vol.5 (1), 65-69 (2014)
2-Total Domination In Graphs V. R. Kulli Department of Mathematics, Gulbarga University, Gulbarga, INDIA. (Received on: February 4, 2014) ABSTRACT A vertex set SV of a graph G=(V, E) is a 2-total dominating set of G if every vertex in V is adjacent to at least 2 vertices in S. The minimum cardinality of a 2-total dominating set is called the 2total domination number of G and is denoted by t2(G). In this paper, we initiate a study of 2-total domination in graphs. Some bounds on t2(G) are obtained and its exact values for some standard graphs are established. Mathematics Subject Classification: 05C. Keywords: total domination, 2-total domination.
1. INTRODUCTION All graphs considered here are finite, nonempty, undirected without loops and multiple edges. We follow the notation and terminology of Harary2. Unless and otherwise stated, the graphs G=(V, E) considered here have p=|V| vertices and q = |E| edges. A set D of vertices in a graph G is called a dominating set if every vertex in V – D is adjacent to some vertex in D. The domination number (G) of G is the minimum cardinality of a dominating set of G. Recently several domination parameters are given in the books by Kulli3,7,8. A set D of vertices in a graph G is a total dominating
set of G if every vertex in V is adjacent to at least one vertex in D. The total domination number t(G) of G is the minimum cardinality of a total dominating set of G,1. Kulli and Janakiram9 introduced the concept of (n,m)-total domination in graphs. A set D of vertices in a graph G is an (n,m)total dominating set of G if each vertex vV is adjacent to at least n vertices in D and m vertices in V – D. The (n,m)-total domination number tn,m(G) of G is the minimum cardinality of an (n,m)-total dominating set. If n =2, m =2, then tn,m(G)=t2, 2(G). Kulli5 studied the concept of (2,2)-total domination in graphs. In a graph G, a vertex is said to dominate itself
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V. R. Kulli, J. Comp. & Math. Sci. Vol.5 (1), 65-69 (2014)
and all its neighbors. Kulli3 defined a slight different concept called (n,m)-tuple domination. A set D of vertices in a graph G is an (n,m)-tuple dominating set if each vertex in V is dominated by at least n vertices of D and at least m vertices of V – D. The (n,m)-tuple domination number ×n,m(G) is the minimum cardinality of an (n, m)- tuple dominating set of G. If n = 2, m = 2, then ×n,m(G) = ×2,2(G). In this case ×2,2(G) is called the (2,2)-tuple domination number of G. Kulli and Sigarkanti10 introduced the concept of (n,m)-domination in graphs. A set D of vertices in G is an (n,m)-dominating set if every vertex u V – D, |N(u) D|n and for every vertex v D, |N(v) (V –D)|m. The (n, m)domination number n,m(G) of G is the minimum cardinality of an (n,m)-dominating set of G. Kulli6 studied the concept of (2, 2)domination in graphs. Kulli4 introduced the concept of ntotal domination as follows: A nonempty set D of vertices in a graph G is an n-total dominating set if every vertex in V is adjacent to at least n vertices in D. The n-total domination number tn(G) of G is the minimum cardinality of an n-total dominating set of G. If n = 2, then tn(G)=t2(G). In this case t2(G) is called the 2-total domination number (or double total domination number) of G,3. A t2-set is a minimum 2-total dominating set. Let (G) denote the maximum degree of G. The open neighborhood of a vertex v in G is the set N(v) = {uV : uvE}.
2. RESULTS Remark 1. The 2-total domination number t2(G) is only defined for graphs with minimum degree at least 2. Proposition 2. If t2(G) exists, deg v 2 for every vertex v in G.
then
Proof: From Remark 1, the result follows. Proposition 3. For any graph G with an endvertex, t2(G) does not exist. Proof: From Remark 1, the result follows. Corollary 4. For any tree T, t2(T) does not exist. Proposition 5. For any graph G with t2-set, t(G) t2(G). (1) Proof: Every 2-total dominating set of G is a total dominating set of G. Thus (1) holds. Theorem 6. If G is a connected graph with minimum degree 2 such that all induced subgraphs HG have minimum degree less than 2, then t2(G) = p. Proof: Suppose G satisfies the condition of Theorem 6. Suppose t2(G) < p and D is a minimum 2-total dominating set of G. Then there exists a vertex vi in V – D and the degree of each vertex in D is at least 2. Then clearly the induced subgraph Ddoes not contain a vertex of degree less than 2, a contradiction. Thus vi D and t2(G) = p.
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We obtain the exact values of t2(G) for some standard graphs. Proposition 7[4]. For any cycle Cp with p 3 vertices, t2(Cp)=p. Proof: This follows from Theorem 6. Proposition 8. For any 2-regular graph G with p vertices, t2(G)=p. Proof: This follows from Theorem 6. Proposition 9. For any complete graph Kp with p3 vertices, t2(Kp)=3. Proof: If D is a t2- set of Kp, then every vertex in V is adjacent with at least 2 vertices of D. Hence if uD, then u is adjacent with at least 2 vertices of D. Thus D has at least 3 vertices. Thus since D is a minimum 2-total dominating set, t2(Kp)=3. Proposition 10. For any complete bipartite graph Km,n with 2mn, t2(Km,n) = 4. Proof: Let Km, n be the complete bipartite graph on vertex sets V1 and V2 such that |V1| = m and |V2| = n. Let D be a t2-set of Km,n. Suppose viV1. Then vi is adjacent with at least 2 vertices of V2. Suppose vjV2. Then vj is adjacent with at least 2 vertices of V1. Thus a 2-total dominating set contains at least 4 vertices of Km,n. Thus since D is a t2set,
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t2(Km, n) = 4. Proposition 11[4]. For any wheel Wp with p 5 vertices, t2(Wp) = p – 2. From the above results, the following theorem gives an upper bound for 2-total domination number. Theorem 12. If G is a connected graph with p 3 vertices and (G)=2, then t2(G) p and this bound is sharp. The cycles Cp with p3 vertices achieve the upper bound. The lower bound (1) can be improved slightly. Theorem 13. If G is a connected graph with (G) 2, then t2(G) t(G) + 1 and this bound is sharp. Proof: Suppose D is a t2-set of G. We consider the following 2 cases: Case 1. Suppose uV–D. Let u be adjacent with at least 2 vertices of D, say v1, v2, .... Suppose S=D –{v1}. Therefore since D is a 2-total dominating set, u is adjacent with at least one vertex of S. Case 2. Suppose uD. Let u be adjacent with at least 2 vertices of D, say v1, v2, ... Suppose S = D – {v1}. Therefore since D is a 2-total dominating set, u is adjacent with at least one vertex of S. From the above two cases, we see that every vertex of G is adjacent with at least one vertex of S. Hence S is a total dominating set. Thus
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t(G)|S| = |D| – |{v1}|= t2(G) – 1. Therefore t2(G) t(G) + 1. The complete graphs Kp with p3 vertices achieve this bound. The following theorem is due to Kulli in4. Theorem 14[4]. (i) If G has p vertices and minimum degree n and t(G) = p – (G) + 1, then tn(G) p – (G) + n. (ii) If G is connected with minimum degree n, (G) < p – 1 and t(G) = p – (G), tn(G) p – (G) + n – 1. The next result gives lower bounds on t2(G) for certain graphs. Theorem 15. (i) If G has p vertices with (G)=2 and t(G)=p – (G)+1, then p – (G)+2 t2(G). (2) (ii) If G is connected with (G)=2, (G) < p – 1 and t(G) = p – (G), then p – (G)+1 t2(G). (3) Proof: The bounds (2) and (3) follow from Theorem 14. The graph 2K3 achieves the lower bound (2). Theorem 16. For a connected graph G with p 3 vertices and with t2-set, 3 t2(G) (4) and the bound is sharp. Proof: The lower bound follows from the definition of t2(G). The complete graphs Kp with p3 vertices realize the sharp lower bound.
3. OPEN PROBLEMS The following are some problems for further investigation. Problem 1. Characterize graphs G for which t2(G) = p. Problem 2. Characterize graphs G for which t2(G) = t(G)+1. Problem 3. Characterize graphs G for which t2(G) = p – (G) + 2. Problem 4. Characterize graphs G for which t2(G) = p – (G) + 1. REFERENCES 1. E.J. Cockayne, R.M. Dawes and S.T. Hedetniemi, Total domination in graphs, Networks, 10, 211-219 (1980). 2. F. Harary, Graph Theory, Addison Wesley, Reading Mass. (1969). 3. V.R. Kulli, Theory of Domination in Graphs, Vishwa International Publications, Gulbarga, India (2010). 4. V.R. Kulli, On n-total domination number in graphs. In Graph Theory, Combinatorics¸ Algorithms and Applications, Y. Alavi, et al. (eds.), SIAM Philadelphia, 319-324 (1991). 5. V.R. Kulli, (2, 2)-total domination in graphs, manuscript, (2013). 6. V.R. Kulli, (2,2)-domination in graphs, manuscript, (2013). 7. V.R. Kulli, Advances in Domination Theory I, Vishwa International Publications, Gulbarga, India (2012). 8. V.R. Kulli, Advances in Domination Theory II, Vishwa International Publications, Gulbarga, India (2013).
Journal of Computer and Mathematical Sciences Vol. 5, Issue 1, 28 February, 2014 Pages (1-122)
V. R. Kulli, J. Comp. & Math. Sci. Vol.5 (1), 65-69 (2014)
9. V. R. Kulli and B. Janakiram, The (n,m)-total domination number of a graph, Nat. Acad. Sci. Lett., 24, 132-136 (2001).
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10. V.R. Kulli and S.C. Sigarkanti, The nmdomination number of a graph, Journal of Interdisciplinary Mathematics, 3, 191194 (2000).
Journal of Computer and Mathematical Sciences Vol. 5, Issue 1, 28 February, 2014 Pages (1-122)