signed total roman k-domination in directed graphs
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abstract
let $d$ be a finite and simple digraph with vertex set $v(d)$.a signed total roman $k$-dominating function (str$k$df) on$d$ is a function $f:v(d)rightarrow{-1, 1, 2}$ satisfying the conditionsthat (i) $sum_{xin n^{-}(v)}f(x)ge k$ for each$vin v(d)$, where $n^{-}(v)$ consists of all vertices of $d$ fromwhich arcs go into $v$, and (ii) every vertex $u$ for which$f(u)=-1$ has an inner neighbor $v$ for which $f(v)=2$.the weight of an str$k$df $f$ is $omega(f)=sum_{vin v (d)}f(v)$.the signed total roman $k$-domination number $gamma^{k}_{str}(d)$of $d$ is the minimum weight of an str$k$df on $d$. in this paper weinitiate the study of the signed total roman $k$-domination numberof digraphs, and we present different bounds on $gamma^{k}_{str}(d)$.in addition, we determine the signed total roman $k$-dominationnumber of some classes of digraphs. some of our results are extensionsof known properties of the signed total roman $k$-dominationnumber $gamma^{k}_{str}(g)$ of graphs $g$.
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Journal title:
communication in combinatorics and optimizationجلد ۱، شماره ۲، صفحات ۱۶۵-۱۷۸
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