On the algorithmic complexity of edge total domination

Let G=(V,E)G=(V,E) be a graph with vertex set V and edge set E  . A subset F⊆EF⊆E is an edge total dominating set   if every edge e∈Ee∈E is adjacent to at least one edge in F. The edge total domination problem is to find a minimum edge total dominating set of G. In the present paper, we prove that the edge total domination problem is NP-complete for planar graphs with maximum degree three, and for undirected path graphs, a subclass of chordal graphs and a superclass of trees. We also design a linear-time algorithm for solving this problem in a tree.