  An efficient dominating set (or perfect code) in a graph is a set of vertices
the closed neighborhoods of which partition the vertex set of the graph. The
minimum weight efficient domination problem is the problem of finding an
efficient dominating set of minimum weight in a given vertex-weighted graph;
the maximum weight efficient domination problem is defined similarly. We
develop a framework for solving the weighted efficient domination problems
based on a reduction to the maximum weight independent set problem in the
square of the input graph. Using this approach, we improve on several previous
results from the literature by deriving polynomial-time algorithms for the
weighted efficient domination problems in the classes of dually chordal and
AT-free graphs. In particular, this answers a question by Lu and Tang regarding
the complexity of the minimum weight efficient domination problem in strongly
chordal graphs.
