Defending the Roman Empire: a new strategy
Discrete Mathematics - Special issue: The 18th British combinatorial conference
ROMAN DOMINATION: a parameterized perspective
International Journal of Computer Mathematics
Extremal Problems for Roman Domination
SIAM Journal on Discrete Mathematics
Roman domination over some graph classes
WG'05 Proceedings of the 31st international conference on Graph-Theoretic Concepts in Computer Science
Roman domination on strongly chordal graphs
Journal of Combinatorial Optimization
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A Roman dominating function of a graph $G$ is a function $f$$: V(G) \to \{0, 1, 2\}$ such that whenever $f(v)=0$, there exists a vertex $u$ adjacent to $v$ such that $f(u) = 2$. The weight of $f$ is $w(f) = \sum_{v \in V(G)} f(v)$. The Roman domination number $\gamma_R(G)$ of $G$ is the minimum weight of a Roman dominating function of $G$. Chambers, Kinnersley, Prince, and West [SIAM J. Discrete Math., 23 (2009), pp. 1575-1586] conjectured that $\gamma_R(G) \le \lceil 2n/3 \rceil$ for any $2$-connected graph $G$ of $n$ vertices. This paper gives counterexamples to the conjecture and proves that $\gamma_R(G) \le \max\{\lceil 2n/3 \rceil, 23n/34\}$ for any $2$-connected graph $G$ of $n$ vertices. We also characterize $2$-connected graphs $G$ for which $\gamma_R(G) = 23n/34$ when $23n/34 \lceil 2n/3 \rceil$.