  A graph is an efficient open (resp.\ closed) domination graph if there exists
a subset of vertices whose open (resp.\ closed) neighborhoods partition its
vertex set. Graphs that are efficient open as well as efficient closed (shortly
EOCD graphs) are investigated. The structure of EOCD graphs with respect to
their efficient open and efficient closed dominating sets is explained. It is
shown that the decision problem regarding whether a graph is an EOCD graph is
an NP-complete problem. A recursive description that constructs all EOCD trees
is given and EOCD graphs are characterized among the Sierpi\'nski graphs.
