# The variety of domination games

Boštjan Brešar, Csilla Bujtás, Tanja Gologranc, Sandi Klavžar, Gašper Košmrlj, Tilen Marc, Balázs Patkós, Z. Tuza, Máté Vizer

Research output: Contribution to journalArticle

1 Citation (Scopus)

### Abstract

Domination game (Brešar et al. in SIAM J Discrete Math 24:979–991, 2010) and total domination game (Henning et al. in Graphs Comb 31:1453–1462 (2015) are by now well established games played on graphs by two players, named Dominator and Staller. In this paper, Z-domination game, L-domination game, and LL-domination game are introduced as natural companions of the standard domination games. Versions of the Continuation Principle are proved for the new games. It is proved that in each of these games the outcome of the game, which is a corresponding graph invariant, differs by at most one depending whether Dominator or Staller starts the game. The hierarchy of the five domination games is established. The invariants are also bounded with respect to the (total) domination number and to the order of a graph. Values of the three new invariants are determined for paths up to a small constant independent from the length of a path. Several open problems and a conjecture are listed. The latter asserts that the L-domination game number is not greater than 6 / 7 of the order of a graph.

Original language English Aequationes Mathematicae https://doi.org/10.1007/s00010-019-00661-w Published - Jan 1 2019

### Fingerprint

Domination
Game
Graph in graph theory
Total Domination
Total Domination number
Graph Invariants
Path
Invariant
Continuation
Open Problems

### Keywords

• Domination game
• Grundy domination number
• L-domination game
• Total domination game
• Z-domination game

### ASJC Scopus subject areas

• Mathematics(all)
• Discrete Mathematics and Combinatorics
• Applied Mathematics

### Cite this

Brešar, B., Bujtás, C., Gologranc, T., Klavžar, S., Košmrlj, G., Marc, T., ... Vizer, M. (2019). The variety of domination games. Aequationes Mathematicae. https://doi.org/10.1007/s00010-019-00661-w

The variety of domination games. / Brešar, Boštjan; Bujtás, Csilla; Gologranc, Tanja; Klavžar, Sandi; Košmrlj, Gašper; Marc, Tilen; Patkós, Balázs; Tuza, Z.; Vizer, Máté.

In: Aequationes Mathematicae, 01.01.2019.

Research output: Contribution to journalArticle

Brešar, B, Bujtás, C, Gologranc, T, Klavžar, S, Košmrlj, G, Marc, T, Patkós, B, Tuza, Z & Vizer, M 2019, 'The variety of domination games', Aequationes Mathematicae. https://doi.org/10.1007/s00010-019-00661-w
Brešar B, Bujtás C, Gologranc T, Klavžar S, Košmrlj G, Marc T et al. The variety of domination games. Aequationes Mathematicae. 2019 Jan 1. https://doi.org/10.1007/s00010-019-00661-w
Brešar, Boštjan ; Bujtás, Csilla ; Gologranc, Tanja ; Klavžar, Sandi ; Košmrlj, Gašper ; Marc, Tilen ; Patkós, Balázs ; Tuza, Z. ; Vizer, Máté. / The variety of domination games. In: Aequationes Mathematicae. 2019.
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