Article in volume
Authors:
Lutz Volkmann
Lehrstuhl II für Mathematik, RWTH-Aachen52056 Aachen GERMANY
email: volkm@math2.rwth-aachen.de
Title:
Double Roman and double Italian domination
PDFSource:
Discussiones Mathematicae Graph Theory 43(3) (2023) 721-730
Received: 2020-10-21 , Revised: 2021-02-10 , Accepted: 2021-02-15 , Available online: 2021-03-11 , https://doi.org/10.7151/dmgt.2399
Abstract:
Let $G$ be a graph with vertex set $V(G)$. A double Roman dominating
function (DRDF) on a graph $G$ is a function $f:V(G)\longrightarrow\{0,1,2,3\}$
that satisfies the following conditions: (i) If $f(v)=0$, then $v$ must have a
neighbor $w$ with $f(w)=3$ or two neighbors $x$ and $y$ with $f(x)=f(y)=2$;
(ii) If $f(v)=1$, then $v$ must have a neighbor $w$ with $f(w)\ge 2$. The
weight of a DRDF $f$ is the sum $\sum_{v\in V(G)}f(v)$. The double Roman
domination number equals the minimum weight of a double Roman dominating
function on $G$. A double Italian dominating function (DIDF) is a
function $f:V(G)\longrightarrow \{0,1,2,3\}$ having the property that
$f(N[u])\geq 3$ for every vertex $u\in V(G)$ with $f(u)\in \{0,1\}$, where
$N[u]$ is the closed neighborhood of $v$. The weight of a DIDF $f$ is the sum
$\sum_{v\in V(G)}f(v)$, and the minimum weight of a DIDF in a graph $G$ is the
double Italian domination number. In this paper we first present
Nordhaus-Gaddum type bounds on the double Roman domination number which improved
corresponding results given in [N. Jafari Rad and H. Rahbani, Some progress
on the double Roman domination in graphs, Discuss. Math. Graph Theory 39
(2019) 41–53]. Furthermore, we establish lower bounds on the double Roman
and double Italian domination numbers of trees.
Keywords:
double Roman domination, double Italian domination
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