A subset \(S\) of the vertex set \(V(G)\) of a graph \(G\) is called dominating in \(G\), if each vertex of \(G\) either is in \(S\), or is adjacent to a vertex of \(S\). The minimum number of vertices of a dominating set in \(G\) is the dominating number \(\gamma(G)\) of \(G\). The minimum number of vertices of a set which is dominating in \(G\) and induces a connected subgraph of \(G\) is the connected domination number \(\gamma_c(G)\) of \(G\). The paper studies graphs \(G\) for which \(\gamma(G)= \gamma_c(G)\). Trees and unicyclic graphs with this property are characterized. There are only five cubic graphs having this property. They are listed.