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Total Domination Multisubdivision Number of a Graph

Diana Avella-Alaminos; Magda Dettlaff; Magdalena Lemańska; Rita Zuazua

Discussiones Mathematicae Graph Theory (2015)

  • Volume: 35, Issue: 2, page 315-327
  • ISSN: 2083-5892

Abstract

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The domination multisubdivision number of a nonempty graph G was defined in [3] as the minimum positive integer k such that there exists an edge which must be subdivided k times to increase the domination number of G. Similarly we define the total domination multisubdivision number msdγt (G) of a graph G and we show that for any connected graph G of order at least two, msdγt (G) ≤ 3. We show that for trees the total domination multisubdi- vision number is equal to the known total domination subdivision number. We also determine the total domination multisubdivision number for some classes of graphs and characterize trees T with msdγt (T) = 1.

How to cite

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Diana Avella-Alaminos, et al. "Total Domination Multisubdivision Number of a Graph." Discussiones Mathematicae Graph Theory 35.2 (2015): 315-327. <http://eudml.org/doc/271098>.

@article{DianaAvella2015,
abstract = {The domination multisubdivision number of a nonempty graph G was defined in [3] as the minimum positive integer k such that there exists an edge which must be subdivided k times to increase the domination number of G. Similarly we define the total domination multisubdivision number msdγt (G) of a graph G and we show that for any connected graph G of order at least two, msdγt (G) ≤ 3. We show that for trees the total domination multisubdi- vision number is equal to the known total domination subdivision number. We also determine the total domination multisubdivision number for some classes of graphs and characterize trees T with msdγt (T) = 1.},
author = {Diana Avella-Alaminos, Magda Dettlaff, Magdalena Lemańska, Rita Zuazua},
journal = {Discussiones Mathematicae Graph Theory},
keywords = {(total) domination; (total) domination subdivision number; (total) domination multisubdivision number; trees; total domination; total domination subdivision number; total domination multisubdivision number},
language = {eng},
number = {2},
pages = {315-327},
title = {Total Domination Multisubdivision Number of a Graph},
url = {http://eudml.org/doc/271098},
volume = {35},
year = {2015},
}

TY - JOUR
AU - Diana Avella-Alaminos
AU - Magda Dettlaff
AU - Magdalena Lemańska
AU - Rita Zuazua
TI - Total Domination Multisubdivision Number of a Graph
JO - Discussiones Mathematicae Graph Theory
PY - 2015
VL - 35
IS - 2
SP - 315
EP - 327
AB - The domination multisubdivision number of a nonempty graph G was defined in [3] as the minimum positive integer k such that there exists an edge which must be subdivided k times to increase the domination number of G. Similarly we define the total domination multisubdivision number msdγt (G) of a graph G and we show that for any connected graph G of order at least two, msdγt (G) ≤ 3. We show that for trees the total domination multisubdi- vision number is equal to the known total domination subdivision number. We also determine the total domination multisubdivision number for some classes of graphs and characterize trees T with msdγt (T) = 1.
LA - eng
KW - (total) domination; (total) domination subdivision number; (total) domination multisubdivision number; trees; total domination; total domination subdivision number; total domination multisubdivision number
UR - http://eudml.org/doc/271098
ER -

References

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  1. [1] H. Aram, S.M. Sheikholeslami and O. Favaron, Domination subdivision number of trees, Discrete Math. 309 (2009) 622-628. doi:10.1016/j.disc.2007.12.085[Crossref] Zbl1171.05380
  2. [2] S. Benecke and C.M. Mynhardt, Trees with domination subdivision number one, Australas. J. Combin. 42 (2008) 201-209. Zbl1153.05018
  3. [3] M. Dettlaff, J. Raczek and J. Topp, Domination subdivision and multisubdivision numbers of graphs, submitted. 
  4. [4] O. Favaron, H. Karami and S.M. Sheikholeslami, Disproof of a conjecture on the subdivision domination number of a graph, Graphs Combin. 24 (2008) 309-312. doi:10.1007/s00373-008-0788-6[Crossref] Zbl1193.05124
  5. [5] T.W. Haynes, S.T. Hedetniemi and P.J. Slater, Fundamentals of Domination in Graphs (Marcel Dekker Inc., New York, 1998). Zbl0890.05002
  6. [6] T.W. Haynes, S.T. Hedetniemi and L.C. van der Merwe, Total domination subdivi- sion numbers, J. Combin. Math. Combin. Comput. 44 (2003) 115-128. Zbl1020.05048
  7. [7] T.W. Haynes, M.A. Henning and L.S. Hopkins, Total domination subdivision num- bers of graphs, Discuss. Math. Graph Theory 24 (2004) 457-467. doi:10.7151/dmgt.1244[Crossref] 
  8. [8] T.W. Haynes, M.A. Henning and L.S. Hopkins, Total domination subdivision num- bers of trees, Discrete Math. 286 (2004) 195-202. doi:10.1016/j.disc.2004.06.004[Crossref] 
  9. [9] H. Karami, A. Khodkar, R. Khoeilar and S.M. Sheikholeslami, Trees whose total domination subdivision number is one, Bull. Inst. Combin. Appl. 53 (2008) 56-57. Zbl1168.05050
  10. [10] S. Velammal, Studies in Graph Theory: Covering, Independence, Domination and Related Topics, Ph.D. Thesis (Manonmaniam Sundaranar University, Tirunelveli, 1997). 

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