The -rainbow bondage number of a graph

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• 作者：N. Dehgardi^a ; S.M. Sheikholeslami^a ; ^{s.m.sheikholeslami@azaruniv.edu" class="auth_mail} ; ^{sm_sheikholeslami@yahoo.com" class="auth_mail} ; L. Volkmann^b ; ^{volkm@math2.rwth-aachen.de" class="auth_mail}
• 关键词：kk-rainbow dominating function ; kk-rainbow domination number ; kk-rainbow bondage number
• 刊名：Discrete Applied Mathematics
• 出版年：10 September 2014
• 年：2014
• 卷：174
• 期：Complete
• 页码：133-139
• 全文大小：400 K

文摘

For a positive integer X1400208X&_mathId=si2.gif&_user=111111111&_pii=S0166218X1400208X&_rdoc=1&_issn=0166218X&md5=12e1aadfb97eacf8de366bd1b0521278" title="Click to view the MathML source">kmi>kmi>, a kmi>kmi>-rainbow dominating function   (kRDF) of a graph Gmi>Gmi> is a function fmi>fmi> from the vertex set V(G)mi>Vmi>(<mi>Gmi>) to the set of all subsets of the set {1,2,…,k}mi>kmi>} such that for any vertex 142d7d8ecbd54fbca1417f56567c" title="Click to view the MathML source">v∈V(G)mi>vmi>∈<mi>Vmi>(<mi>Gmi>) with f(v)=0谈mi>fmi>(<mi>vmi>)=0谈 the condition 鈰?sub>u∈N(v)f(u)={1,2,…,k}mi>umi>∈<mi>Nmi>(<mi>vmi>)<mi>fmi>(<mi>umi>)={1,2,…,<mi>kmi>} is fulfilled, where N(v)mi>Nmi>(<mi>vmi>) is the neighborhood of vmi>vmi>. The weight   of a kRDF fmi>fmi> is the value 蠅(f)=∑_v∈V|f(v)|mi>蠅mi>(<mi>fmi>)=∑<mi>vmi>∈<mi>Vmi>|<mi>fmi>(<mi>vmi>)|. The kmi>kmi>-rainbow domination number   of a graph Gmi>Gmi>, denoted by 纬_rk(G)mi>纬mi><mi>rmi><mi>kmi>(<mi>Gmi>), is the minimum weight of a kRDF of G. The 1-rainbow domination is the same as the ordinary domination. The kmi>kmi>-rainbow bondage number b_rk(G)mi>bmi><mi>rmi><mi>kmi>(<mi>Gmi>) of a graph Gmi>Gmi> with maximum degree at least two is the minimum cardinality of all sets E^′⊆E(G)mi>Emi>′⊆<mi>Emi>(<mi>Gmi>) for which 纬_rk(G&minus;E^′)>纬_rk(G)mi>纬mi><mi>rmi><mi>kmi>(<mi>Gmi>&minus;<mi>Emi>′)><mi>纬mi><mi>rmi><mi>kmi>(<mi>Gmi>). Note that b_r1(G)mi>bmi><mi>rmi>1(<mi>Gmi>) is the classical bondage number b(G)mi>bmi>(<mi>Gmi>). In this paper, we initiate the study of the kmi>kmi>-rainbow bondage number in graphs and we present some bounds for b_rk(G)mi>bmi><mi>rmi><mi>kmi>(<mi>Gmi>). In addition, we determine the 2-rainbow bondage number of some classes of graphs.