A Roman dominating function on a graph \(G = (V, E)\) is a function \(f : V \to \{0,1,2\}\) satisfying the condition that every vertex \(u \in V\) for which \(f(u) = 0\) is adjacent to a vertex \(v\) for which \(f(v) = 2\). The weight of a Roman dominating function is the value \(f(V) = \sum_{u \in V} f(u)\). The Roman domination number, \(\gamma_R(G)\), of \(G\) is the minimum weight of a Roman dominating function on \(G\). In this paper, we study those graphs for which the removal of any pair of vertices decreases the Roman domination number. A graph \(G\) is said to be \emph{Roman domination bicritical} or just \(\gamma_R\)-bicritical, if \(\gamma_R(G – \{v,u\}) < \gamma_R(G)\) for any pair of vertices \(v,u \in V\). We study properties of \(\gamma_R\)-bicritical graphs, and we characterize \(\gamma_R\)-bicritical trees and unicyclic graphs.