Signed Double Roman Domination of Graphs

Abstract

In this paper we continue the study of signed double Roman dominating
functions in graphs. A signed double Roman dominating function (SDRDF) on a
graph $G=(V,E)$ is a function $f:V(G)\rightarrow\{-1,1,2,3\}$ having the
property that for each $v\in V(G)$, $f[v]\geq1$, and if $f(v)=-1$, then vertex
$v$ has at least two neighbors assigned 2 under $f$ or one neighbor $w$ with
$f(w)=3$, and if $f(v)=1$, then vertex $v$ must have at leat one neighbor $w$
with $f(w)\geq2$. The weight of a SDRDF is the sum of its function values over
all vertices. The signed double Roman domination number $\gamma_{sdR}(G)$ is
the minimum weight of a SDRDF on $G$. We present several lower bounds on the
signed double Roman domination number of a graph in terms of various graph
invariants. In particular, we show that if $G$ is a graph of order $n$ and
size $m$ with no isolated vertex, then $\gamma_{sdR}(G)\geq\frac{19n-24m}{9}$
and $\gamma_{sdR}(G)\geq4\sqrt{\frac{n}{3}}-n.$ Moreover, we characterize the
graphs attaining equality in these two bounds.