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

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 Sierpiński graphs.