Graphs and Combinatorics

, Volume 31, Issue 5, pp 1453–1462

# Total Version of the Domination Game

• Michael A. Henning
• Sandi Klavžar
• Douglas F. Rall
Original Paper

## Abstract

In this paper, we continue the study of the domination game in graphs introduced by Brešar et al. (SIAM J Discret Math 24:979–991, 2010). We study the total version of the domination game and show that these two versions differ significantly. We present a key lemma, known as the Total Continuation Principle, to compare the Dominator-start total domination game and the Staller-start total domination game. Relationships between the game total domination number and the total domination number, as well as between the game total domination number and the domination number, are established.

## Keywords

Domination game Total domination number

## Mathematics Subject Classification

05C57 91A43 05C69

## Notes

### Acknowledgments

Research supported in part by the South African National Research Foundation and the University of Johannesburg, by the Ministry of Science of Slovenia under the Grants P1-0297, and by a Grant from the Simons Foundation (#209654 to Douglas Rall) and by the Wylie Enrichment Fund of Furman University. We thank the referees for several useful suggestions.

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## Authors and Affiliations

• Michael A. Henning
• 1
• Sandi Klavžar
• 2
• 3
• 4
• Douglas F. Rall
• 5
1. 1.Department of MathematicsUniversity of JohannesburgJohannesburgSouth Africa
2. 2.Faculty of Mathematics and PhysicsUniversity of LjubljanaLjubljanaSlovenia
3. 3.Faculty of Natural Sciences and MathematicsUniversity of MariborMariborSlovenia
4. 4.Institute of Mathematics, Physics and MechanicsLjubljanaSlovenia
5. 5.Department of MathematicsFurman UniversityGreenvilleUSA