Global Roman domination in graphs

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• 作者：P. Roushini Leely Pushpam^a ; ^{roushinip@yahoo.com" class="auth_mail" title="E-mail the corresponding author} ; S. Padmapriea^b ; ^{prieas@gmail.com" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Roman domination ; Global domination
• 刊名：Discrete Applied Mathematics
• 出版年：2016
• 出版时间：19 February 2016
• 年：2016
• 卷：200
• 期：Complete
• 页码：176-185
• 全文大小：487 K

文摘

A Roman dominating function   (RDF) on a graph hmlsrc">hImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0166218X15003558&_mathId=si4.gif&_user=111111111&_pii=S0166218X15003558&_rdoc=1&_issn=0166218X&md5=81c0a202ac6ebcd5694b35a09ad52704" title="Click to view the MathML source">G=(V,E)hContainer hidden">hCode">h altimg="si4.gif" overflow="scroll">G=(V,E)h> is defined to be a function hmlsrc">hImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0166218X15003558&_mathId=si5.gif&_user=111111111&_pii=S0166218X15003558&_rdoc=1&_issn=0166218X&md5=d6f6f7ba203117cd7d8199ce87abcf78" title="Click to view the MathML source">f:V→{0,1,2}hContainer hidden">hCode">h altimg="si5.gif" overflow="scroll">f:V→{0,1,2}h> satisfying the condition that every vertex hmlsrc">hImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0166218X15003558&_mathId=si6.gif&_user=111111111&_pii=S0166218X15003558&_rdoc=1&_issn=0166218X&md5=44b7cc65b8c96a69b55024b8d19aae91" title="Click to view the MathML source">uhContainer hidden">hCode">h altimg="si6.gif" overflow="scroll">uh> for which hmlsrc">hImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0166218X15003558&_mathId=si7.gif&_user=111111111&_pii=S0166218X15003558&_rdoc=1&_issn=0166218X&md5=5d67326aa7ef29385df30b2f030942eb" title="Click to view the MathML source">f(u)=0hContainer hidden">hCode">h altimg="si7.gif" overflow="scroll">f(u)=0h> is adjacent to at least one vertex hmlsrc">hImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0166218X15003558&_mathId=si8.gif&_user=111111111&_pii=S0166218X15003558&_rdoc=1&_issn=0166218X&md5=6876b348fe80a4212a2e84907421e850" title="Click to view the MathML source">vhContainer hidden">hCode">h altimg="si8.gif" overflow="scroll">vh> for which hmlsrc">hImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0166218X15003558&_mathId=si9.gif&_user=111111111&_pii=S0166218X15003558&_rdoc=1&_issn=0166218X&md5=eb451f6315aea6c814814f1458462a95" title="Click to view the MathML source">f(v)=2hContainer hidden">hCode">h altimg="si9.gif" overflow="scroll">f(v)=2h>. A set hmlsrc">hImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0166218X15003558&_mathId=si10.gif&_user=111111111&_pii=S0166218X15003558&_rdoc=1&_issn=0166218X&md5=bb811237d994e1ed65bab3bef142a9dd" title="Click to view the MathML source">S⊆VhContainer hidden">hCode">h altimg="si10.gif" overflow="scroll">S⊆Vh> is a global dominating set   if hmlsrc">hImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0166218X15003558&_mathId=si11.gif&_user=111111111&_pii=S0166218X15003558&_rdoc=1&_issn=0166218X&md5=3416c9eb74bb316ec74b51f4c8a71b26" title="Click to view the MathML source">ShContainer hidden">hCode">h altimg="si11.gif" overflow="scroll">Sh> dominates both hmlsrc">hImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0166218X15003558&_mathId=si12.gif&_user=111111111&_pii=S0166218X15003558&_rdoc=1&_issn=0166218X&md5=a80dd8bda91cb3ee6401c5ceb35e2b26" title="Click to view the MathML source">GhContainer hidden">hCode">h altimg="si12.gif" overflow="scroll">Gh> and its complement hmlsrc">he MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0166218X15003558&_mathId=si13.gif&_user=111111111&_pii=S0166218X15003558&_rdoc=1&_issn=0166218X&md5=159b8209558763c0a5a34137a4357d27">height="14" width="12" alt="View the MathML source" style="margin-top: -5px; vertical-align: middle" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0166218X15003558-si13.gif">hContainer hidden">hCode">h altimg="si13.gif" overflow="scroll">G¯h>. The global domination number hmlsrc">hImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0166218X15003558&_mathId=si14.gif&_user=111111111&_pii=S0166218X15003558&_rdoc=1&_issn=0166218X&md5=855b6a0236bc7ca1efd3bc1ce84e5ebb" title="Click to view the MathML source">γ_g(G)hContainer hidden">hCode">h altimg="si14.gif" overflow="scroll">γg(G)h> of a graph hmlsrc">hImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0166218X15003558&_mathId=si12.gif&_user=111111111&_pii=S0166218X15003558&_rdoc=1&_issn=0166218X&md5=a80dd8bda91cb3ee6401c5ceb35e2b26" title="Click to view the MathML source">GhContainer hidden">hCode">h altimg="si12.gif" overflow="scroll">Gh> is the minimum cardinality of hmlsrc">hImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0166218X15003558&_mathId=si11.gif&_user=111111111&_pii=S0166218X15003558&_rdoc=1&_issn=0166218X&md5=3416c9eb74bb316ec74b51f4c8a71b26" title="Click to view the MathML source">ShContainer hidden">hCode">h altimg="si11.gif" overflow="scroll">Sh>. We define a global Roman dominating function   on a graph hmlsrc">hImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0166218X15003558&_mathId=si4.gif&_user=111111111&_pii=S0166218X15003558&_rdoc=1&_issn=0166218X&md5=81c0a202ac6ebcd5694b35a09ad52704" title="Click to view the MathML source">G=(V,E)hContainer hidden">hCode">h altimg="si4.gif" overflow="scroll">G=(V,E)h> to be a function hmlsrc">hImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0166218X15003558&_mathId=si5.gif&_user=111111111&_pii=S0166218X15003558&_rdoc=1&_issn=0166218X&md5=d6f6f7ba203117cd7d8199ce87abcf78" title="Click to view the MathML source">f:V→{0,1,2}hContainer hidden">hCode">h altimg="si5.gif" overflow="scroll">f:V→{0,1,2}h> such that hmlsrc">hImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0166218X15003558&_mathId=si19.gif&_user=111111111&_pii=S0166218X15003558&_rdoc=1&_issn=0166218X&md5=decb52adb16379f6d811f775e71330ea" title="Click to view the MathML source">fhContainer hidden">hCode">h altimg="si19.gif" overflow="scroll">fh> is an RDF for both hmlsrc">hImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0166218X15003558&_mathId=si12.gif&_user=111111111&_pii=S0166218X15003558&_rdoc=1&_issn=0166218X&md5=a80dd8bda91cb3ee6401c5ceb35e2b26" title="Click to view the MathML source">GhContainer hidden">hCode">h altimg="si12.gif" overflow="scroll">Gh> and its complement hmlsrc">he MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0166218X15003558&_mathId=si13.gif&_user=111111111&_pii=S0166218X15003558&_rdoc=1&_issn=0166218X&md5=159b8209558763c0a5a34137a4357d27">height="14" width="12" alt="View the MathML source" style="margin-top: -5px; vertical-align: middle" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0166218X15003558-si13.gif">hContainer hidden">hCode">h altimg="si13.gif" overflow="scroll">G¯h>. The weight   of a global Roman dominating function is the value hmlsrc">hImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0166218X15003558&_mathId=si22.gif&_user=111111111&_pii=S0166218X15003558&_rdoc=1&_issn=0166218X&md5=b7e8c02a92de490f50d375f35dbb296d" title="Click to view the MathML source">f(V)=∑_u∈Vf(u)hContainer hidden">hCode">h altimg="si22.gif" overflow="scroll">f(V)=∑u∈Vf(u)h>. The minimum weight of a global Roman dominating function on a graph hmlsrc">hImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0166218X15003558&_mathId=si12.gif&_user=111111111&_pii=S0166218X15003558&_rdoc=1&_issn=0166218X&md5=a80dd8bda91cb3ee6401c5ceb35e2b26" title="Click to view the MathML source">GhContainer hidden">hCode">h altimg="si12.gif" overflow="scroll">Gh> is called the global Roman domination number   of hmlsrc">hImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0166218X15003558&_mathId=si12.gif&_user=111111111&_pii=S0166218X15003558&_rdoc=1&_issn=0166218X&md5=a80dd8bda91cb3ee6401c5ceb35e2b26" title="Click to view the MathML source">GhContainer hidden">hCode">h altimg="si12.gif" overflow="scroll">Gh> and denoted by hmlsrc">hImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0166218X15003558&_mathId=si25.gif&_user=111111111&_pii=S0166218X15003558&_rdoc=1&_issn=0166218X&md5=72b7cbebfc60812515dda88e686f078a" title="Click to view the MathML source">γ_gR(G)hContainer hidden">hCode">h altimg="si25.gif" overflow="scroll">γgR(G)h>. In this paper, we initiate a study of this parameter.