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 $\emph{edge dominating set}$ (resp. a $\emph{total edge dominating set}$) 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 {\em edge domination number} (resp. {\em total edge domination number}) of $G$, denoted by $\gamma^{'}(G)$ (resp. $\gamma_t^{'}(G)$). In the present paper, we prove that the total edge domination problem is NP-complete for bipartite graphs with maximum degree 3. We also design a linear-time algorithm for solving this problem for trees. Finally, for a graph $G$, we give the inequality $\gamma^{'}(G)\leqslant \gamma^{'}_{t}(G)\leqslant 2\gamma^{'}(G)$ and characterize the trees $T$ which obtain the upper or lower bounds in the inequality.