The complexity of total edge domination and some related results on trees

For a graph G = ( V , E ) with vertex set V and edge set E , a subset F of E is called an edge dominating set (resp. a total edge dominating set ) if every edge in E \ F (resp. in E ) is adjacent to at least one edge in F , the minimum cardinality of an edge dominating set (resp. a total edge dominating set) of G is the edge domination number (resp. total edge domination number ) of G , denoted by γ ′ ( G ) (resp. γ t ′ ( G ) ). In the present paper, we first prove that the total edge domination problem is NP-complete for bipartite graphs with maximum degree 3. Then, for a graph G , we give the inequality γ ′ ( G ) ⩽ γ t ′ ( G ) ⩽ 2 γ ′ ( G ) and characterize the trees T which obtain the upper or lower bounds in the inequality.