Advertisement

On two conjectures concerning total domination subdivision number in graphs

  • Rana Khoeilar
  • Hossein Karami
  • Seyed Mahmoud SheikholeslamiEmail author
Article
  • 12 Downloads

Abstract

A subset S of vertices of a graph G without isolated vertex is a total dominating set if every vertex of V(G) is adjacent to some vertex in S. The total domination number\(\gamma _t(G)\) is the minimum cardinality of a total dominating set of G. The total domination subdivision number\(\mathrm{sd}_{\gamma _t}(G)\) is the minimum number of edges that must be subdivided (each edge in G can be subdivided at most once) in order to increase the total domination number. In this paper we prove that for any connected graph G of order \(n\ge 3\), \(\mathrm{sd}_{\gamma _t}(G)\le \gamma _t(G)+1\) and for any connected graph G of order \(n\ge 5\), \(\mathrm{sd}_{\gamma _t}(G)\le \frac{n+1}{2}\), answering two conjectures posed in Favaron et al. (J Comb Optim 20:76–84, 2010a).

Keywords

Matching Barrier Total domination number Total domination subdivision number 

Notes

References

  1. Favaron O, Karami H, Sheikholeslami SM (2007) Total domination and total domination subdivision numbers of graphs. Australas J Comb 38:229–235zbMATHGoogle Scholar
  2. Favaron O, Karami H, Khoeilar R, Sheikholeslami SM (2009) A new upper bound on the total domination subdivision numbers. Graphs Combin 25:41–47MathSciNetCrossRefzbMATHGoogle Scholar
  3. Favaron O, Karami H, Khoeilar R, Sheikholeslami SM (2010a) On the total domination subdivision number in some classes of graphs. J Comb Optim 20:76–84MathSciNetCrossRefzbMATHGoogle Scholar
  4. Favaron O, Karami H, Khoeilar R, Sheikholeslami SM (2010b) Matching and total domination subdivision number of graphs with few \(C_4\). Discuss Math Graph Theory 30:611–618MathSciNetCrossRefzbMATHGoogle Scholar
  5. Favaron O, Karami H, Sheikholeslami SM (2011) Bounding the total domination subdivision number of a graph in terms of its order. J Comb Optim 21:209–218MathSciNetCrossRefzbMATHGoogle Scholar
  6. Haynes TW, Hedetniemi ST, van der Merwe LC (2003) Total domination subdivision numbers. J Comb Math Comb Comput 44:115–128MathSciNetzbMATHGoogle Scholar
  7. Haynes TW, Henning MA, Hopkins LS (2004a) Total domination subdivision numbers of graphs. Discuss Math Gr Theory 24:457–467MathSciNetCrossRefzbMATHGoogle Scholar
  8. Haynes TW, Henning MA, Hopkins LS (2004b) Total domination subdivision numbers of trees. Discrete Math 286:195–202MathSciNetCrossRefzbMATHGoogle Scholar
  9. Karami H, Khodkar A, Khoeilar R, Sheikholeslami SM (2008) Trees whose total domination subdivision number is one. Bull Inst Comb Appl 53:57–67MathSciNetzbMATHGoogle Scholar
  10. Karami H, Khodkar A, Sheikholeslami SM (2011) An upper bound for total domination subdivision numbers of graphs. Ars Comb 102:321–331zbMATHGoogle Scholar
  11. Lovász L, Plummer MD (1986) Matching theory, vol 29. Annals of Discrete Mathematics. North Holland, AmsterdamzbMATHGoogle Scholar
  12. Sheikholeslami SM (2010) On the total domination subdivision number of a graph. Cent Eur J Math 8:468–473MathSciNetCrossRefzbMATHGoogle Scholar
  13. Tutte WT (1947) The factorization of linear graphs. J Lond Math Soc 22:107–111MathSciNetCrossRefzbMATHGoogle Scholar
  14. Velammal S (1997) Studies in graph theory: covering, independence, domination and related topics. Ph.D. thesis, Manonmaniam Sundaranar University, TirunelveliGoogle Scholar
  15. West DB (2000) Introduction to graph theory. Prentice-Hall, Inc., Upper Saddle RiverGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of MathematicsAzarbaijan Shahid Madani UniversityTabrizIslamic Republic of Iran

Personalised recommendations