Iranian Journal of Mathematical Sciences and Informatics
مجله علوم ریاضی و انفورماتیک
IJMSI
Basic Sciences
http://ijmsi.ir
1
admin
1735-4463
2008-9473
8
10.61882/ijmsi
14
8888
13
en
jalali
1399
7
1
gregorian
2020
10
1
15
2
online
1
fulltext
en
Roman k-Tuple Domination in Graphs
تخصصي
Special
پژوهشي
Research paper
<pre style="margin-top: 0px; margin-bottom: 0px;">
<span style=" color:#000000;">For any integer </span><span style=" color:#008000;">$kgeq 1$</span><span style=" color:#000000;"> and any graph </span><span style=" color:#008000;">$G=(V,E)$</span><span style=" color:#000000;"> with minimum degree at least </span><span style=" color:#008000;">$k-1$</span>‎<span style=" color:#000000;">, </span>‎<span style=" color:#000000;">we define a</span>‎ ‎<span color:="" sans="" source="" style="font-family: Tahoma, Arial, Verdana, ">function </span><span color:="" sans="" source="" style="font-family: Tahoma, Arial, Verdana, ">$f:Vrightarrow {0,1,2}$</span><span color:="" sans="" source="" style="font-family: Tahoma, Arial, Verdana, "> as a Roman </span><span color:="" sans="" source="" style="font-family: Tahoma, Arial, Verdana, ">$k$</span><span color:="" sans="" source="" style="font-family: Tahoma, Arial, Verdana, ">-tuple dominating</span><span sans="" source="" style="font-family: Tahoma, Arial, Verdana, ">‎ </span>‎<span color:="" sans="" source="" style="font-family: Tahoma, Arial, Verdana, ">function on </span><span color:="" sans="" source="" style="font-family: Tahoma, Arial, Verdana, ">$G$</span><span color:="" sans="" source="" style="font-family: Tahoma, Arial, Verdana, "> if for any vertex </span><span color:="" sans="" source="" style="font-family: Tahoma, Arial, Verdana, ">$v$</span><span color:="" sans="" source="" style="font-family: Tahoma, Arial, Verdana, "> with </span><span color:="" sans="" source="" style="font-family: Tahoma, Arial, Verdana, ">$f(v)=0$</span><span color:="" sans="" source="" style="font-family: Tahoma, Arial, Verdana, "> there exist at least</span><span sans="" source="" style="font-family: Tahoma, Arial, Verdana, ">‎ </span>‎<span color:="" sans="" source="" style="font-family: Tahoma, Arial, Verdana, ">$k$</span><span color:="" sans="" source="" style="font-family: Tahoma, Arial, Verdana, "> and for any vertex </span><span color:="" sans="" source="" style="font-family: Tahoma, Arial, Verdana, ">$v$</span><span color:="" sans="" source="" style="font-family: Tahoma, Arial, Verdana, "> with </span><span color:="" sans="" source="" style="font-family: Tahoma, Arial, Verdana, ">$f(v)neq 0$</span><span color:="" sans="" source="" style="font-family: Tahoma, Arial, Verdana, "> at least </span><span color:="" sans="" source="" style="font-family: Tahoma, Arial, Verdana, ">$k-1$</span><span color:="" sans="" source="" style="font-family: Tahoma, Arial, Verdana, "> vertices in its neighborhood with </span><span color:="" sans="" source="" style="font-family: Tahoma, Arial, Verdana, ">$f(w)=2$</span><span sans="" source="" style="font-family: Tahoma, Arial, Verdana, ">‎</span><span color:="" sans="" source="" style="font-family: Tahoma, Arial, Verdana, ">. </span><span sans="" source="" style="font-family: Tahoma, Arial, Verdana, ">‎</span><span color:="" sans="" source="" style="font-family: Tahoma, Arial, Verdana, ">The minimum weight of a Roman </span><span color:="" sans="" source="" style="font-family: Tahoma, Arial, Verdana, ">$k$</span><span color:="" sans="" source="" style="font-family: Tahoma, Arial, Verdana, ">-tuple dominating function </span><span color:="" sans="" source="" style="font-family: Tahoma, Arial, Verdana, ">$f$</span><span color:="" sans="" source="" style="font-family: Tahoma, Arial, Verdana, "> on </span><span color:="" sans="" source="" style="font-family: Tahoma, Arial, Verdana, ">$G$</span><span color:="" sans="" source="" style="font-family: Tahoma, Arial, Verdana, "> is called the Roman </span><span color:="" sans="" source="" style="font-family: Tahoma, Arial, Verdana, ">$k$</span><span color:="" sans="" source="" style="font-family: Tahoma, Arial, Verdana, ">-tuple domination number of the graph where the weight of </span><span color:="" sans="" source="" style="font-family: Tahoma, Arial, Verdana, ">$f$</span><span color:="" sans="" source="" style="font-family: Tahoma, Arial, Verdana, "> is </span><span color:="" sans="" source="" style="font-family: Tahoma, Arial, Verdana, ">$f(V)=sum_{vin V}f(v)$</span><span sans="" source="" style="font-family: Tahoma, Arial, Verdana, ">‎</span><span color:="" sans="" source="" style="font-family: Tahoma, Arial, Verdana, ">. </span></pre>
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<pre style="margin-top: 0px; margin-bottom: 0px;">
‎<span style=" color:#000000;">In this paper</span>‎<span style=" color:#000000;">, </span>‎<span style=" color:#000000;">we initiate to study the Roman </span><span style=" color:#008000;">$k$</span><span style=" color:#000000;">-tuple</span>‎ ‎<span color:="" sans="" source="" style="font-family: Tahoma, Arial, Verdana, ">domination number of a graph</span><span sans="" source="" style="font-family: Tahoma, Arial, Verdana, ">‎</span><span color:="" sans="" source="" style="font-family: Tahoma, Arial, Verdana, ">, </span><span sans="" source="" style="font-family: Tahoma, Arial, Verdana, ">‎</span><span color:="" sans="" source="" style="font-family: Tahoma, Arial, Verdana, ">by giving some sharp bounds for the Roman </span><span color:="" sans="" source="" style="font-family: Tahoma, Arial, Verdana, ">$k$</span><span color:="" sans="" source="" style="font-family: Tahoma, Arial, Verdana, ">-tuple domination number of a garph</span><span sans="" source="" style="font-family: Tahoma, Arial, Verdana, ">‎</span><span color:="" sans="" source="" style="font-family: Tahoma, Arial, Verdana, ">, </span><span sans="" source="" style="font-family: Tahoma, Arial, Verdana, ">‎</span><span color:="" sans="" source="" style="font-family: Tahoma, Arial, Verdana, ">the Mycieleskian of a graph</span><span sans="" source="" style="font-family: Tahoma, Arial, Verdana, ">‎</span><span color:="" sans="" source="" style="font-family: Tahoma, Arial, Verdana, ">, </span><span sans="" source="" style="font-family: Tahoma, Arial, Verdana, ">‎</span><span color:="" sans="" source="" style="font-family: Tahoma, Arial, Verdana, ">and the corona graphs</span><span sans="" source="" style="font-family: Tahoma, Arial, Verdana, ">‎</span><span color:="" sans="" source="" style="font-family: Tahoma, Arial, Verdana, ">. </span><span sans="" source="" style="font-family: Tahoma, Arial, Verdana, ">‎</span><span color:="" sans="" source="" style="font-family: Tahoma, Arial, Verdana, ">Also finding the Roman </span><span color:="" sans="" source="" style="font-family: Tahoma, Arial, Verdana, ">$k$</span><span color:="" sans="" source="" style="font-family: Tahoma, Arial, Verdana, ">-tuple domination number of some known graphs is our other goal</span><span sans="" source="" style="font-family: Tahoma, Arial, Verdana, ">‎</span><span color:="" sans="" source="" style="font-family: Tahoma, Arial, Verdana, ">. </span><span sans="" source="" style="font-family: Tahoma, Arial, Verdana, ">‎</span><span color:="" sans="" source="" style="font-family: Tahoma, Arial, Verdana, ">Some of our results extend these one</span><span sans="" source="" style="font-family: Tahoma, Arial, Verdana, ">‎ </span>‎<span color:="" sans="" source="" style="font-family: Tahoma, Arial, Verdana, ">given by Cockayne and et al</span><span sans="" source="" style="font-family: Tahoma, Arial, Verdana, ">‎</span><span color:="" sans="" source="" style="font-family: Tahoma, Arial, Verdana, ">. </span><span sans="" source="" style="font-family: Tahoma, Arial, Verdana, ">‎</span><span color:="" sans="" source="" style="font-family: Tahoma, Arial, Verdana, ">cite</span><span color:="" sans="" source="" style="font-family: Tahoma, Arial, Verdana, ">{CDHH04} in 2004 for the Roman</span><span sans="" source="" style="font-family: Tahoma, Arial, Verdana, ">‎ </span>‎<span color:="" sans="" source="" style="font-family: Tahoma, Arial, Verdana, ">domination number</span><span sans="" source="" style="font-family: Tahoma, Arial, Verdana, ">‎</span><span color:="" sans="" source="" style="font-family: Tahoma, Arial, Verdana, ">.</span></pre>
Roman $k$-tuple domination number, Roman $k$-tuple graph, $k$-Tuple domination number, $k$-Tuple total domination number, Mycieleskian of a graph.
101
115
http://ijmsi.ir/browse.php?a_code=A-10-140-3&slc_lang=en&sid=1
A.
P. Kazemi
adelpkazemi@yahoo.com
10031947532846008758
10031947532846008758
Yes
U. of Mohaghegh Ardabili