# Weak roman domination in graphs

• Volume: 31, Issue: 1, page 161-170
• ISSN: 2083-5892

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## Abstract

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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 $w\left(f\right)={\sum }_{v\in V}f\left(v\right)$. The weak Roman domination number, denoted by ${\gamma }_{r}\left(G\right)$, is the minimum weight of a WRDF in G. In this paper, we characterize the class of trees and split graphs for which ${\gamma }_{r}\left(G\right)=\gamma \left(G\right)$ and find ${\gamma }_{r}$-value for a caterpillar, a 2×n grid graph and a complete binary tree.

## How to cite

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P. Roushini Leely Pushpam, and T.N.M. Malini Mai. "Weak roman domination in graphs." Discussiones Mathematicae Graph Theory 31.1 (2011): 161-170. <http://eudml.org/doc/270829>.

@article{P2011,
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 $w(f) = ∑_\{v ∈ V\}f(v)$. The weak Roman domination number, denoted by $γ_r(G)$, is the minimum weight of a WRDF in G. In this paper, we characterize the class of trees and split graphs for which $γ_r(G) = γ(G)$ and find $γ_r$-value for a caterpillar, a 2×n grid graph and a complete binary tree.},
author = {P. Roushini Leely Pushpam, T.N.M. Malini Mai},
journal = {Discussiones Mathematicae Graph Theory},
keywords = {domination number; weak Roman domination number},
language = {eng},
number = {1},
pages = {161-170},
title = {Weak roman domination in graphs},
url = {http://eudml.org/doc/270829},
volume = {31},
year = {2011},
}

TY - JOUR
AU - P. Roushini Leely Pushpam
AU - T.N.M. Malini Mai
TI - Weak roman domination in graphs
JO - Discussiones Mathematicae Graph Theory
PY - 2011
VL - 31
IS - 1
SP - 161
EP - 170
AB - 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 $w(f) = ∑_{v ∈ V}f(v)$. The weak Roman domination number, denoted by $γ_r(G)$, is the minimum weight of a WRDF in G. In this paper, we characterize the class of trees and split graphs for which $γ_r(G) = γ(G)$ and find $γ_r$-value for a caterpillar, a 2×n grid graph and a complete binary tree.
LA - eng
KW - domination number; weak Roman domination number
UR - http://eudml.org/doc/270829
ER -

## References

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10. [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. Zbl1189.05138
11. [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. Zbl1195.05056
12. [12] I. Stewart, Defend the Roman Empire, Scientific American 281 (1999) 136-139.

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