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

Abstract

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 \(E\backslash F\) (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 \(\gamma ^{\prime }(G)\) (resp. \(\gamma _t^{\prime }(G)\)). 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 \(\gamma ^{\prime }(G)\leqslant \gamma ^{\prime }_{t}(G)\leqslant 2\gamma ^{\prime }(G)\) and characterize the trees T which obtain the upper or lower bounds in the inequality.