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A graph is an efficient open (resp.\ closed) domination graph if there existsa subset of vertices whose open (resp.\ closed) neighborhoods partition itsvertex set. Graphs that are efficient open as well as efficient closed (shortlyEOCD graphs) are investigated. The structure of EOCD graphs with respect totheir efficient open and efficient closed dominating sets is explained. It isshown that the decision problem regarding whether a graph is an EOCD graph isan NP-complete problem. A recursive description that constructs all EOCD treesis given and EOCD graphs are characterized among the Sierpi\'nski graphs.

Graphs that are simultaneously efficient open domination and efficient closed domination graphs