Extremal Problems for Roman Domination

Authors

E. W. Chambers
W. Kinnersley
N. Prince, Illinois Mathematics and Science AcademyFollow
D. B. West

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 ∑_ʋϵV(G)f(v) over such functions. Let G be a connected n-vertex graph. We prove that γ_R(G) ≤ 4n/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) ≤ 8n/11 when ᵟ(G) ≥ 2 and n ≥ 9, and this is sharp.

Comments

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

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.