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## Discussiones Mathematicae Graph Theory

2011 | 31 | 1 | 161-170
Tytuł artykułu

### Weak roman domination in graphs

Autorzy
Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
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.
Słowa kluczowe
EN
Wydawca
Czasopismo
Rocznik
Tom
Numer
Strony
161-170
Opis fizyczny
Daty
wydano
2011
otrzymano
2009-11-07
poprawiono
2010-04-02
zaakceptowano
2010-04-06
Twórcy
• D.B. Jain College, Chennai - 600 097, Tamil Nadu, India
autor
• SRR Engineering College, Chennai - 603 103, Tamil Nadu, India
Bibliografia
• [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.
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