  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.
