Results on the maximal Roman domination number in graphs
Subject Areas : StatisticsMaryam Kamalipashakolaee 1 , Hossein Abdollahzadeh Ahangar 2 , Mehran Motiee 3 , Seyed Mahmoud Sheikholeslami 4
1 - Department of Mathematics, Babol Noshirvani University of Technology, Babol, Iran
2 - Department of Mathematics, Babol Noshirvani University of Technology, Babol, Iran
3 - Department of Mathematics, Babol Noshirvani University of Technology, Babol, Iran
4 - Department of MathematicsAzarbaijan Shahid Madani UniversityTabriz-Iran
Keywords: عدد احاطهگر رومی, تابع احاطهگر رومی ماکسیمال, تابع احاطهگر رومی, عدد احاطهگر رومی ماکسیمال,
Abstract :
A Roman dominating function on a graph G is a labeling f:V(G)→{0,1,2} such that every vertex with label 0 has a neighbor with label 2. A Roman dominating function on a graph G is a labeling f:V(G)→{0,1,2} such that every vertex with label 0 has a neighbor with label 2. A maximal Roman dominating function on a graph G is a Roman dominating function f such that V_0={w ∈V(G)│f(w)=0} is not a dominating set of G. The weight of maximal Roman dominating function is the value w(f)=f(V(G))=∑_(x∈V(G))▒〖f(x).〗 The maximal Roman dominating number γ_mR (G) of a graph G equals the minimum weight of a maximal Roman dominating function on G. In this paper, we continue the study of maximal Roman domination number. First, we characterize all graphs G of order n with g(G)≥6 for which γ_mR (G) =n-2, and then, we consider this property for some graphs with girth at most 5.
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