On the semitotal domination number of line graphs

Abstract Given a graph G containing no isolated vertex, a set S ⊆ V ( G ) is said to be a semitotal dominating set of G if for every vertex in V ( G ) ∖ S , there is a vertex in S adjacent to it; and for every vertex in S there is another vertex in S within a distance of 2. The semitotal domination number of G , denoted as γ t 2 ( G ) , is the minimum cardinality of a semitotal dominating set of G . In this work, we consider two conjectures (proposed in Goddard et al. (2014) and Zhu et al. (2017)) concerning this parameter by showing their validity for line graphs. Moreover, we show that it is NP-complete to determine γ t 2 ( L ( G ) ) for a planar graph G with maximum degree 4, where L ( G ) denotes the line graphs of G .