A set D of vertices in a graph G is an efficient dominating set (e.d.s. for short) of G if D is an independent set and every vertex not in D is adjacent to exactly one vertex in D. The efficient domination (ED) problem asks for the existence of an e.d.s. in G. The minimum weighted efficient domination problem (MIN-WED for short) is the problem of finding an e.d.s. of minimum weight in a given vertex-weighted graph. Brandstädt, Fičur, Leitert and Milanič (2015) [3] stated the running times of the fastest known polynomial-time algorithms for the MIN-WED problem on some graphs classes by using a Hasse diagram.In this paper, we update this Hasse diagram by showing that, while for every integer d such that d=3k or d=3k+2, where k≥1, the ED problem remains NP-complete for graphs of diameter d, the weighted version of the problem is solvable in time O(|V(G)|+|E(G)|) in the class of diameter three bipartite graphs and in time O(|V(G)|5) in the class of diameter three planar graphs.