If \(D\) is a dominating set in a graph \(G\) and \(G[D]\) is connected, then \(D\) is a connected dominating set. The minimum size of a connected dominating set in \(G\) is called the connected domination number of \(G\), denoted by \(\gamma_c(G)\). A graph \(G\) is a perfect connected-dominant graph if \(\gamma(H)= \gamma_c(H)\) for each connected induced subgraph \(H\) of \(G\). The author proves that a graph \(G\) is perfect connected-dominant if and only if \(G\) has neigher induced \(P_5\) nor induced \(C_5\).