Discussiones Mathematicae Graph Theory 31(1) (2011) 161-170
doi: 10.7151/dmgt.1532

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WEAK ROMAN DOMINATION IN GRAPHS

P. Roushini Leely Pushpam

D.B. Jain College
Chennai - 600 097, Tamil Nadu, India
e-mail: roushinip@yahoo.com

T.N.M. Malini Mai

SRR Engineering College
Chennai - 603 103, Tamil Nadu, India
e-mail: malinitnm2008@yahoo.com

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.

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Received 7 November 2009
Revised 2 April 2010
Accepted 6 April 2010