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Discussiones Mathematicae Graph Theory 31(1) (2011)
161-170
DOI: https://doi.org/10.7151/dmgt.1532
WEAK ROMAN DOMINATION IN GRAPHS
P. Roushini Leely Pushpam
D.B. Jain College | T.N.M. Malini Mai
SRR Engineering College |
Abstract
Let G = (V,E) be a graph and f be a function f:V→{0,1,2}. A vertex u with f(u) = 0 is said to be undefended with respect to f, if it is not adjacent to a vertex with positive weight. The function f is a weak Roman dominating function (WRDF) if each vertex u with f(u) = 0 is adjacent to a vertex v with f(v) > 0 such that the function f′: V → {0,1,2} defined by f′(u) = 1, f′(v) = f(v)−1 and f′(w) = f(w) if w ∈ V−{u,v}, has no undefended vertex. The weight of f is w(f) = ∑v ∈ Vf(v). The weak Roman domination number, denoted by γr(G), is the minimum weight of a WRDF in G. In this paper, we characterize the class of trees and split graphs for which γr(G) = γ(G) and find γr-value for a caterpillar, a 2 ×n grid graph and a complete binary tree.Keywords: domination number, weak Roman domination number.
2010 Mathematics Subject Classification: 05C.
References
[1] | E.J. Cockayne, P.A. Dreyer, S.M. Hedetniemi and S.T. Hedetniemi, Roman domination in graphs, Discrete Math. 78 (2004) 11-22, doi: 10.1016/j.disc.2003.06.004. |
[2] | T.W. Haynes, S.T. Hedetniemi and P.J. Slater, (Eds), Fundamentals of Domination in Graphs (Marcel Dekker, New York, 1998). |
[3] | T.W. Haynes, S.T. Hedetniemi and P.J. Slater, (Eds), Domination in Graphs; Advanced Topics (Marcel Dekker, Inc. New York, 1998). |
[4] | S.T. Hedetniemi and M.A. Henning, Defending the Roman Empire - A new strategy, Discrete Math. 266 (2003) 239-251, doi: 10.1016/S0012-365X(02)00811-7. |
[5] | M.A. Henning, A characterization of Roman trees, Discuss. Math. Graph Theory 22 (2002) 325-334, doi: 10.7151/dmgt.1178. |
[6] | M.A. Henning, Defending the Roman Empire from multiple attacks, Discrete Math. 271 (2003) 101-115, doi: 10.1016/S0012-365X(03)00040-2. |
[7] | C.S. ReVelle, Can you protect the Roman Empire?, John Hopkins Magazine (2) (1997) 70. |
[8] | C.S. ReVelle and K.E. Rosing, Defendens Romanum: Imperium problem in military strategy, American Mathematical Monthly 107 (2000) 585-594, doi: 10.2307/2589113. |
[9] | R.R. Rubalcaba and P.J. Slater, Roman Dominating Influence Parameters, Discrete Math. 307 (2007) 3194-3200, doi: 10.1016/j.disc.2007.03.020. |
[10] | P. Roushini Leely Pushpam and T.N.M. Malini Mai, On Efficient Roman dominatable graphs, J. Combin Math. Combin. Comput. 67 (2008) 49-58. |
[11] | P. Roushini Leely Pushpam and T.N.M. Malini Mai, Edge Roman domination in graphs, J. Combin Math. Combin. Comput. 69 (2009) 175-182. |
[12] | I. Stewart, Defend the Roman Empire, Scientific American 281 (1999) 136-139. |
Received 7 November 2009
Revised 2 April 2010
Accepted 6 April 2010
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