### Abstract

Given a graph G, a real-valued function f: V (G) [0; 1] is a fractional dominat-ing function if Pu2 N[v] f(u) > 1 holds for every vertex v and its closed neighborhood N[v] in G. The aim is to minimize the sumPv2V (G)f(v). A different approach to graph domination is the domination game, introducedby Breffsar et al. [SIAM J. Discrete Math. 24 (2010) 979[991]. It is played on agraph G by two players, namely Dominator and Staller, who take turns choosing avertex such that at least one previously undominated vertex becomes dominated.The game is over when all vertices are dominated. Dominator wants to finish thegame as soon as possible, while Staller wants to delay the end. Assuming thatboth players play optimally and Dominator starts, the length of the game on G isuniquely determined and is called the game domination number of G. We introduce and study the fractional version of the domination game, wherethe moves are ruled by the condition of fractional domination. Here we prove afundamental property of this new game, namely the fractional version of the so-called Continuation Principle. Moreover we present lower and upper bounds on thefractional game domination number of paths and cycles. These estimates are tightapart from a small additive constant. We also prove that the game dominationnumber cannot be bounded above by any linear function of the fractional game domination number.

Original language | English |
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Article number | P4.3 |

Journal | Electronic Journal of Combinatorics |

Volume | 26 |

Issue number | 4 |

Publication status | Published - Jan 1 2019 |

### ASJC Scopus subject areas

- Theoretical Computer Science
- Geometry and Topology
- Discrete Mathematics and Combinatorics
- Computational Theory and Mathematics
- Applied Mathematics

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## Cite this

*Electronic Journal of Combinatorics*,

*26*(4), [P4.3].