Geodesic
The Semitotal Domination Problem in Block Graphs
Discussiones Mathematicae. Graph Theory, Tome 42 (2022) no. 1, pp. 231-248

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A set D of vertices in a graph G is a dominating set of G if every vertex outside D is adjacent in G to some vertex in D. A set D of vertices in G is a semitotal dominating set of G if D is a dominating set of G and every vertex in D is within distance 2 from another vertex of D. Given a graph G and a positive integer k, the semitotal domination problem is to decide whether G has a semitotal dominating set of cardinality at most k. The semitotal domination problem is known to be NP-complete for chordal graphs and bipartite graphs as shown in [M.A. Henning and A. Pandey, Algorithmic aspects of semitotal domination in graphs, Theoret. Comput. Sci. 766 (2019) 46–57]. In this paper, we present a linear time algorithm to compute a minimum semitotal dominating set in block graphs. On the other hand, we show that the semitotal domination problem remains NP-complete for undirected path graphs.
Keywords: domination, semitotal domination, block graphs, undirected path graphs, NP-complete 
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Henning, Michael A.; Pal, Saikat; Pradhan, D. The Semitotal Domination Problem in Block Graphs. Discussiones Mathematicae. Graph Theory, Tome 42 (2022) no. 1, pp. 231-248. http://geodesic.mathdoc.fr/item/DMGT_2022_42_1_a14/