#### Document Type

Article

#### Publication Date

2009

#### Disciplines

Discrete Mathematics and Combinatorics | Mathematics

#### Abstract

A *Roman dominating function *of a graph *G* is a labeling *f: V*(*G*) →{0,1,2} such that every vertex with a label 0 has a neighbor with label 2. The *Roman domination number **γ** _{R}*(

*G*) of

*G*is the minimum of ∑

_{ʋϵ}

_{V(}

_{G}_{)}

*f*(

*v*) over such functions. Let

*G*be a connected

*n*-vertex graph. We prove that

*γ*

*(*

_{R}*G*) ≤ 4

*n*/5, and we characterize the graphs achieving equality. We obtain sharp upper and lower bounds for

*γ*

*(*

_{R}*G*) +

*γ*

_{R}(*Ḡ*

*)*and

*γ*

*(*

_{R}*G*)

*γ*

_{R}(*Ḡ*

*)*, improving known results for domination number. We prove that

*γ*

*(*

_{R}*G*) ≤ 8

*n*/11 when ᵟ(

*G*) ≥ 2 and

*n*≥ 9, and this is sharp.

#### Recommended Citation

Chambers, E.W., Kinnersley, W., Prince, N., & West, D.B. (2009). Extremal problems for Roman domination. *SIAM Journal on Discrete Mathematics,* 23(3), 1575-1586.

## Comments

At the time of publication, Noah Prince was affiliated with the University of Illinois at Urbana-Champaign.