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THE DOMINATION GAME ON SPLIT GRAPHS

Part of: Graph theory Game theory

Published online by Cambridge University Press:  12 November 2018

TIJO JAMES ,
SANDI KLAVŽAR Open the ORCID record for SANDI KLAVŽAR [Opens in a new window]  and
AMBAT VIJAYAKUMAR
TIJO JAMES
Affiliation:
Department of Mathematics, Pavanatma College, Murickassery, India Department of Mathematics, Cochin University of Science and Technology, India email tijojames@gmail.com
SANDI KLAVŽAR*
Affiliation:
Faculty of Mathematics and Physics, University of Ljubljana, Slovenia Faculty of Natural Sciences and Mathematics, University of Maribor, Slovenia Institute of Mathematics, Physics and Mechanics, Ljubljana, Slovenia email sandi.klavzar@fmf.uni-lj.si
AMBAT VIJAYAKUMAR
Affiliation:
Department of Mathematics, Cochin University of Science and Technology, India email vambat@gmail.com
*
✉sandi.klavzar@fmf.uni-lj.si
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Abstract

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We investigate the domination game and the game domination number $\unicode[STIX]{x1D6FE}_{g}$ in the class of split graphs. We prove that $\unicode[STIX]{x1D6FE}_{g}(G)\leq n/2$ for any isolate-free $n$-vertex split graph $G$, thus strengthening the conjectured $3n/5$ general bound and supporting Rall’s $\lceil n/2\rceil$-conjecture. We also characterise split graphs of even order with $\unicode[STIX]{x1D6FE}_{g}(G)=n/2$.

Keywords

domination gamesplit graphHamiltonian path

MSC classification

Primary: 91A43: Games involving graphs
Secondary: 05C57: Games on graphs 05C69: Dominating sets, independent sets, cliques
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 99 , Issue 2 , April 2019 , pp. 327 - 337
Copyright
© 2018 Australian Mathematical Publishing Association Inc. 

Footnotes

The second author acknowledges the financial support from the Slovenian Research Agency (research core funding No. P1-0297 and the project N1-0043 Combinatorial Problems with an Emphasis on Games).

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