# An upper bound on the double Roman domination number

An upper bound on the double Roman domination number A double Roman dominating function (DRDF) on a graph $$G=(V,E)$$ G = ( V , E ) is a function $$f : V \rightarrow \{0, 1, 2, 3\}$$ f : V → { 0 , 1 , 2 , 3 } having the property that if $$f(v) = 0$$ f ( v ) = 0 , then vertex v must have at least two neighbors assigned 2 under f or one neighbor w with $$f(w)=3$$ f ( w ) = 3 , and if $$f(v)=1$$ f ( v ) = 1 , then vertex v must have at least one neighbor w with $$f(w)\ge 2$$ f ( w ) ≥ 2 . The weight of a DRDF f is the value $$f(V) = \sum _{u \in V}f(u)$$ f ( V ) = ∑ u ∈ V f ( u ) . The double Roman domination number $$\gamma _{dR}(G)$$ γ dR ( G ) of a graph G is the minimum weight of a DRDF on G. Beeler et al. (Discrete Appl Math 211:23–29, 2016) observed that every connected graph G having minimum degree at least two satisfies the inequality $$\gamma _{dR}(G)\le \frac{6n}{5}$$ γ dR ( G ) ≤ 6 n 5 and posed the question whether this bound can be improved. In this paper, we settle the question and prove that for any connected graph G of order n with minimum degree at least two, $$\gamma _{dR}(G)\le \frac{8n}{7}$$ γ dR ( G ) ≤ 8 n 7 . http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Journal of Combinatorial Optimization Springer Journals

# An upper bound on the double Roman domination number

, Volume 36 (1) – Apr 2, 2018
9 pages

/lp/springer_journal/an-upper-bound-on-the-double-roman-domination-number-sR52rHRro7
Publisher
Springer US
Subject
Mathematics; Combinatorics; Convex and Discrete Geometry; Mathematical Modeling and Industrial Mathematics; Theory of Computation; Optimization; Operations Research/Decision Theory
ISSN
1382-6905
eISSN
1573-2886
D.O.I.
10.1007/s10878-018-0286-6
Publisher site
See Article on Publisher Site

### Abstract

A double Roman dominating function (DRDF) on a graph $$G=(V,E)$$ G = ( V , E ) is a function $$f : V \rightarrow \{0, 1, 2, 3\}$$ f : V → { 0 , 1 , 2 , 3 } having the property that if $$f(v) = 0$$ f ( v ) = 0 , then vertex v must have at least two neighbors assigned 2 under f or one neighbor w with $$f(w)=3$$ f ( w ) = 3 , and if $$f(v)=1$$ f ( v ) = 1 , then vertex v must have at least one neighbor w with $$f(w)\ge 2$$ f ( w ) ≥ 2 . The weight of a DRDF f is the value $$f(V) = \sum _{u \in V}f(u)$$ f ( V ) = ∑ u ∈ V f ( u ) . The double Roman domination number $$\gamma _{dR}(G)$$ γ dR ( G ) of a graph G is the minimum weight of a DRDF on G. Beeler et al. (Discrete Appl Math 211:23–29, 2016) observed that every connected graph G having minimum degree at least two satisfies the inequality $$\gamma _{dR}(G)\le \frac{6n}{5}$$ γ dR ( G ) ≤ 6 n 5 and posed the question whether this bound can be improved. In this paper, we settle the question and prove that for any connected graph G of order n with minimum degree at least two, $$\gamma _{dR}(G)\le \frac{8n}{7}$$ γ dR ( G ) ≤ 8 n 7 .

### Journal

Journal of Combinatorial OptimizationSpringer Journals

Published: Apr 2, 2018

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