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

Voir la notice de l'article provenant de la source Library of Science

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 
@article{DMGT_2022_42_1_a14,
     author = {Henning, Michael A. and Pal, Saikat and Pradhan, D.},
     title = {The {Semitotal} {Domination} {Problem} in {Block} {Graphs}},
     journal = {Discussiones Mathematicae. Graph Theory},
     pages = {231--248},
     publisher = {mathdoc},
     volume = {42},
     number = {1},
     year = {2022},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/DMGT_2022_42_1_a14/}
}
TY  - JOUR
AU  - Henning, Michael A.
AU  - Pal, Saikat
AU  - Pradhan, D.
TI  - The Semitotal Domination Problem in Block Graphs
JO  - Discussiones Mathematicae. Graph Theory
PY  - 2022
SP  - 231
EP  - 248
VL  - 42
IS  - 1
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/DMGT_2022_42_1_a14/
LA  - en
ID  - DMGT_2022_42_1_a14
ER  - 
%0 Journal Article
%A Henning, Michael A.
%A Pal, Saikat
%A Pradhan, D.
%T The Semitotal Domination Problem in Block Graphs
%J Discussiones Mathematicae. Graph Theory
%D 2022
%P 231-248
%V 42
%N 1
%I mathdoc
%U http://geodesic.mathdoc.fr/item/DMGT_2022_42_1_a14/
%G en
%F DMGT_2022_42_1_a14
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/

[1] A.V. Aho, J.E. Hopcroft and J.D. Ullman, The Design and Analysis of Computer Algorithms (Addison-Wesley, Boston, 1974).

[2] K.S. Booth and J.H. Johnson, Dominating sets in chordal graphs, SIAM J. Comput. 11 (1982) 191–199. https://doi.org/10.1137/0211015

[3] W. Goddard, M.A. Henning and C.A. McPillan, Semitotal domination in graphs, Util. Math. 94 (2014) 67–81.

[4] T.W. Haynes, S.T. Hedetniemi and P.J. Slater, Fundamentals of Domination in Graphs (Marcel Dekker Inc., New York, 1998).

[5] T.W. Haynes, S.T. Hedetniemi and P.J. Slater, Domination in Graphs: Advanced Topics (Marcel Dekker Inc., New York, 1998).

[6] T.W. Haynes and M.A. Henning, Perfect graphs involving semitotal and semipaired domination, J. Comb. Optim. 36 (2018) 416–433. https://doi.org/10.1007/s10878-018-0303-9

[7] M.A. Henning, Edge weighting functions on semitotal dominating sets, Graphs Combin. 33 (2017) 403–417. https://doi.org/10.1007/s00373-017-1769-4

[8] M.A. Henning and A.J. Marcon, On matching and semitotal domination in graphs, Discrete Math. 324 (2014) 13–18. https://doi.org/10.1016/j.disc.2014.01.021

[9] M.A. Henning and A.J. Marcon, Vertices contained in all or in no minimum semi-total dominating set of a tree, Discuss. Math. Graph Theory 36 (2016) 71–93. https://doi.org/10.7151/dmgt.1844

[10] M.A. Henning and A.J. Marcon, Semitotal domination in claw-free cubic graphs, Ann. Comb. 20 (2016)) 799–813. https://doi.org/10.1007/s00026-016-0331-z

[11] M.A. Henning and A.J. Marcon, Semitotal domination in graphs: Partition and algorithmic results, Util. Math. 106 (2018) 165–184.

[12] M.A. Henning and A. Pandey, Algorithmic aspects of semitotal domination in graphs, Theoret. Comput. Sci. 766 (2019) 46–57. https://doi.org/10.1016/j.tcs.2018.09.019

[13] M.A. Henning and A. Yeo, Total Domination in Graphs (Springer, New York, 2013). https://doi.org/10.1007/978-1-4614-6525-6

[14] Z. Shao and P. Wu, Complexity and approximation ratio of semitotal domination in graphs, Commun. Comb. Optim. 3 (2018) 143–150. https://doi.org/10.22049/CCO.2018.25987.1065

[15] W. Zhuang and G. Hao, Semitotal domination in trees, Discrete Math. Theoret. Comput. Sci. 20 (2) (2018) #5. https://doi.org/10.23638/DMTCS-20-2-5