Complexity and characterization aspects of edge-related domination for graphs

For a connected graph $$G = (V, E)$$ G = ( V , E ) , a subset F of E is an edge dominating set (resp. a total edge dominating set) if every edge in $$E-F$$ 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 $$\gamma '(G)$$ γ ′ ( G ) (resp. $$\gamma '_t(G)$$ γ t ′ ( G ) ). In the present paper, we study a parameter, called the semitotal edge domination number, which is squeezed between $$\gamma '(G)$$ γ ′ ( G ) and $$\gamma '_t(G)$$ γ 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 $$e'$$ e ′ in S that e either is adjacent to $$e'$$ e ′ or shares a common neighbor edge with $$e'$$ e ′ . The semitotal edge domination number, denoted by $$\gamma ^{'}_{st}(G)$$ γ 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 $$\gamma ^{'}(G)=\gamma ^{'}_{st}(G)$$ γ ′ ( G ) = γ st ′ ( G ) or $$\gamma _t^{'}(G)=\gamma ^{'}(G)$$ γ t ′ ( G ) = γ ′ ( 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).