As an additive weight version of the Harary index, the reciprocal degree distance of a simple connected graph \(G\) is defined as \(RDD(G) = \sum\limits_{u,v \subseteq V(G)} \frac{d_G(u)+d_G(v)}{d_G(u,v)}\), where \(d_G(u)\) is the degree of \(u\) and \(d_G(u,v)\) is the distance between \(u\) and \(v\) in \(G\). In this paper, we respectively characterize the extremal graphs with the maximum \(RDD\)-value among all the graphs of order \(n\) with given number of cut vertices and cut edges. In addition, an upper bound on the reciprocal degree distance in terms of the number of cut edges is provided.