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Maker–Breaker Domination Number

Maker–Breaker Domination Number The Maker–Breaker domination game is played on a graph G by Dominator and Staller. The players alternatively select a vertex of G that was not yet chosen in the course of the game. Dominator wins if at some point the vertices he has chosen form a dominating set. Staller wins if Dominator cannot form a dominating set. In this paper, we introduce the Maker–Breaker domination number $$\gamma _{\mathrm{MB}}(G)$$ γ MB ( G ) of G as the minimum number of moves of Dominator to win the game provided that he has a winning strategy and is the first to play. If Staller plays first, then the corresponding invariant is denoted $$\gamma _{\mathrm{MB}}'(G)$$ γ MB ′ ( G ) . Comparing the two invariants, it turns out that they behave much differently than the related game domination numbers. The invariant $$\gamma _{\mathrm{MB}}(G)$$ γ MB ( G ) is also compared with the domination number. Using the Erdős-Selfridge criterion, a large class of graphs G is found for which $$\gamma _{\mathrm{MB}}(G) > \gamma (G)$$ γ MB ( G ) > γ ( G ) holds. Residual graphs are introduced and used to bound/determine $$\gamma _{\mathrm{MB}}(G)$$ γ MB ( G ) and $$\gamma _{\mathrm{MB}}'(G)$$ γ MB ′ ( G ) . Using residual graphs, $$\gamma _{\mathrm{MB}}(T)$$ γ MB ( T ) and $$\gamma _{\mathrm{MB}}'(T)$$ γ MB ′ ( T ) are determined for an arbitrary tree. The invariants are also obtained for cycles and bounded for union of graphs. A list of open problems and directions for further investigations is given. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Bulletin of the Malaysian Mathematical Sciences Society Springer Journals

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References (35)

Publisher
Springer Journals
Copyright
Copyright © 2019 by Malaysian Mathematical Sciences Society and Penerbit Universiti Sains Malaysia
Subject
Mathematics; Mathematics, general; Applications of Mathematics
ISSN
0126-6705
eISSN
2180-4206
DOI
10.1007/s40840-019-00757-1
Publisher site
See Article on Publisher Site

Abstract

The Maker–Breaker domination game is played on a graph G by Dominator and Staller. The players alternatively select a vertex of G that was not yet chosen in the course of the game. Dominator wins if at some point the vertices he has chosen form a dominating set. Staller wins if Dominator cannot form a dominating set. In this paper, we introduce the Maker–Breaker domination number $$\gamma _{\mathrm{MB}}(G)$$ γ MB ( G ) of G as the minimum number of moves of Dominator to win the game provided that he has a winning strategy and is the first to play. If Staller plays first, then the corresponding invariant is denoted $$\gamma _{\mathrm{MB}}'(G)$$ γ MB ′ ( G ) . Comparing the two invariants, it turns out that they behave much differently than the related game domination numbers. The invariant $$\gamma _{\mathrm{MB}}(G)$$ γ MB ( G ) is also compared with the domination number. Using the Erdős-Selfridge criterion, a large class of graphs G is found for which $$\gamma _{\mathrm{MB}}(G) > \gamma (G)$$ γ MB ( G ) > γ ( G ) holds. Residual graphs are introduced and used to bound/determine $$\gamma _{\mathrm{MB}}(G)$$ γ MB ( G ) and $$\gamma _{\mathrm{MB}}'(G)$$ γ MB ′ ( G ) . Using residual graphs, $$\gamma _{\mathrm{MB}}(T)$$ γ MB ( T ) and $$\gamma _{\mathrm{MB}}'(T)$$ γ MB ′ ( T ) are determined for an arbitrary tree. The invariants are also obtained for cycles and bounded for union of graphs. A list of open problems and directions for further investigations is given.

Journal

Bulletin of the Malaysian Mathematical Sciences SocietySpringer Journals

Published: Apr 5, 2019

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