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.

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总边缘控制的复杂性以及树上的一些相关结果

对于具有顶点集V和边集E的图\（G =（V，E）\），如果\（E中的每个边都为E，则E的子集F被称为边占优集（分别是总边占优集）。\反斜杠˚F\） （在RESP。ë）相邻于至少一个边缘˚F的，边缘控制集的最小基数（分别总边缘支配集）ģ是边缘控制数（分别为总边缘G的控制数），用\（\ gamma ^ {\ prime}（G）\）表示（分别\（\ gamma _t ^ {\ prime}（G）\））。在本文中，我们首先证明最大边数为3的二部图的总边控制问题是NP完全的。然后，对于图G，我们给出不等式\（\ gamma ^ {\ prime}（G）\ leqslant \ gamma ^ {\ prime} _ {t}（G）\ leqslant 2 \ gamma ^ {\ prime}（G）\）并刻画得到不等式上界或下界的树T。