For a connected graph \(G = (V, E)\), a subset F of E is 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 \(\gamma '(G)\) (resp. \(\gamma '_t(G)\)). In the present paper, we study a parameter, called the semitotal edge domination number, which is squeezed between \(\gamma '(G)\) and \(\gamma '_t(G)\). A semitotal edge dominating set is an edge dominating set S such that, for every edge e in S, there exists such an edge \(e'\) in S that e either is adjacent to \(e'\) or shares a common neighbor edge with \(e'\). The semitotal edge domination number, denoted by \(\gamma ^{'}_{st}(G)\), is the minimum cardinality of a semitotal edge dominating set of G. In this paper, we prove that the problem of deciding whether \(\gamma ^{'}(G)=\gamma ^{'}_{st}(G)\) or \(\gamma _t^{'}(G)=\gamma ^{'}(G)\) is NP-hard even when restricted to planar graphs with maximum degree 4. We also characterize trees with equal edge domination and semitotal edge domination numbers (Pan et al. in The complexity of total edge domination and some related results on trees, J Comb Optim, 2020, https://doi.org/10.1007/s10878-020-00596-y, we characterized trees with equal edge domination and total edge domination numbers).

中文翻译：

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图的边缘相关支配的复杂性和特征方面

用于连通图\（G =（V，E）\） ，子集˚F的ë是一个边缘支配集（分别为一个总边缘支配集）如果在每个边缘\（EF \） （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（分别为总边控制数的）ģ，记\（\伽马'（G）\）（分别为\（\伽马'_t（G）\））。在本文中，我们研究了一个称为半总边支配数的参数，该参数挤压在\（\ gamma'（G）\）和\（\ gamma'_t（G）\）之间。甲semitotal边缘支配集是边缘控制集š这样，对于每一个边缘ê在小号，存在这样的边缘“（\ \ E）在小号该ë要么邻近\（E” \）或共享公共\（e'\）的邻居边缘。所述semitotal边控制数，记为\（\伽马^ {'} _ {ST}（G）\）是半总边控制集G的最小基数。在本文中，我们证明了确定\（\ gamma ^ {'}（G）= \ gamma ^ {'} _ {st}（G）\）或\（\ gamma _t ^ {'}（即使限制在最大阶数为4的平面图上，G）= \ gamma ^ {'}（G）\）也是NP难的。我们还对边缘支配数和半总边缘支配数相等的树进行了特征描述（Pan等。树上的总边缘控制和一些相关结果，J Comb Optim，2020，https://doi.org/10.1007/s10878-020-00596-y，我们对具有相同边缘控制和总边缘控制数的树进行了特征化）。