﻿ A characterization relating domination, semitotal domination and total Roman domination in trees

# A characterization relating domination, semitotal domination and total Roman domination in trees

##### چکیده

A total Roman dominating function on a graph \$G\$ is a function \$f: V(G) rightarrow {0,1,2}\$ such that for every vertex \$vin V(G)\$ with \$f(v)=0\$ there exists a vertex \$uin V(G)\$ adjacent to \$v\$ with \$f(u)=2\$, and the subgraph induced by the set \${xin V(G): f(x)geq 1}\$ has no isolated vertices. The total Roman domination number of \$G\$, denoted \$gamma_{tR}(G)\$, is the minimum weight \$omega(f)=sum_{vin V(G)}f(v)\$ among all total Roman dominating functions \$f\$ on \$G\$.It is known that \$gamma_{tR}(G)geq gamma_{t2}(G)+gamma(G)\$ for any graph \$G\$ with neither isolated vertex nor components isomorphic to \$K_2\$, where \$gamma_{t2}(G)\$ and \$gamma(G)\$ represent the semitotal domination number and the classical domination number, respectively. In this paper we give a constructive characterization of the trees that satisfy the equality above.

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عنوان ژورنال:

دوره 6  شماره 2

صفحات  197- 209

تاریخ انتشار 2021-12-01

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