Bounds on the outer-independent double Italian domination number

Document Type : Original paper

Authors

1 Shahed University

2 RWTH Aachen University

Abstract

An outer-independent double Italian dominating function (OIDIDF)
on a graph $G$ with vertex set $V(G)$ is a function
$f:V(G)longrightarrow {0,1,2,3}$ such that if $f(v)in{ 0,1}$ for a vertex $vin V(G)$ then $sum_{uin N[v]}f(u)geq3$,
and the set ${uin V(G)|f(u)=0}$ is independent. The weight of
an OIDIDF $f$ is the value $w(f)=sum_{vin V(G)}f(v)$. The
minimum weight of an OIDIDF on a graph $G$ is called the
outer-independent double Italian domination number
$gamma_{oidI}(G)$ of $G$. We present sharp lower bounds for the
outer-independent double Italian domination number of a tree in
terms of diameter, vertex covering number and the order of the
tree.

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