Roman Domination on 2-Connected Graphs

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 $23n/34 \lceil 2n/3 \rceil$.