Trees with Equal 2-domination and 2-independence Numbers
|Mustapha Chellali and Nacéra Meddah |
LAMDA-RO Laboratory, Department of Mathematics
AbstractLet G = (V,E) be a graph. A subset S of V is a 2-dominating set if every vertex of V −S is dominated at least 2 times, and S is a 2-independent set of G if every vertex of S has at most one neighbor in S. The minimum cardinality of a 2-dominating set a of G is the 2-domination number γ2(G) and the maximum cardinality of a 2-independent set of G is the 2-independence number β2(G). Fink and Jacobson proved that γ2(G) ≤ β2(G) for every graph G. In this paper we provide a constructive characterization of trees with equal 2-domination and 2-independence numbers.
Keywords: 2-domination number, 2-independence number, trees
2010 Mathematics Subject Classification: 05C69.
|||M. Borowiecki, On a minimaximal kernel of trees, Discuss. Math. 1 (1975) 3--6.|
|||M. Chellali, O. Favaron, A. Hansberg and L. Volkmann, k-domination and k-independence in graphs: A Survey, Graphs and Combinatorics, 28 (2012) 1--55, doi: 10.1007/s00373-011-1040-3.|
|||O. Favaron, On a conjecture of Fink and Jacobson concerning k-domination and k-dependence, J. Combinat. Theory (B) 39 (1985) 101--102, doi: 10.1016/0095-8956(85)90040-1.|
|||J.F. Fink and M.S. Jacobson, n-domination in graphs, in: Graph Theory with Applications to Algorithms and Computer Science., ed(s), Y. Alavi and A.J. Schwenk (Wiley, New York, 1985) 283--300.|
|||T.W. Haynes, S.T. Hedetniemi and P.J. Slater, Fundamentals of Domination in Graphs ( Marcel Dekker, New York, 1998).|
|||T.W. Haynes, S.T. Hedetniemi and P.J. Slater, Domination in Graphs: Advanced Topics (Marcel Dekker, New York 1998).|
Received 14 September 2010
Revised 10 May 2011
Accepted 11 May 2011