On the efficiency index of a graph

A graph $$G$$ has an efficient dominating set $$D \subseteq V(G)$$ if $$D$$ dominates every vertex exactly once. In this paper we introduce the study of the family $${S_k}$$ of graphs for which every $$G-S$$ is efficiently dominatable for $$0 \le |S|\le k$$ . Assuming that $$G$$ is efficiently dominatable, the efficiency index is the largest value k for which $$G$$ is in $$S_k$$ . A graph $$G$$ will be called super-efficient if every induced subgraph is efficiently dominatable. We give some characterizations for trees, grids, cylinders and torii to be super-efficient.