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This paper deals with the Efficient Edge Domination Problem (EED, for short), also known as Dominating Induced Matching Problem. For an undirected graph G=(V,E) EED asks for an induced matching M⊆E that simultaneously dominates all edges of G. Thus, the distance between edges of M is at least two and every edge in E is adjacent to an edge of M. EED is related to parallel resource allocation problems, encoding theory and network routing. The problem is $\mathbb{NP}$-complete even for restricted classes like planar bipartite and bipartite graphs with maximum degree three. However, the complexity has been open for chordal bipartite graphs. This paper shows that EED can be solved in polynomial time on hole-free graphs. Moreover, it provides even linear time for chordal bipartite graphs. Finally, we strengthen the $\mathbb{NP}$-completeness result to planar bipartite graphs of maximum degree three.