Domination game played on trees and spanning subgraphs
Bostjan Bresar, Sandi Klavzar, Douglas F. Rall

TL;DR
This paper studies the game domination number on trees and spanning subgraphs, establishing bounds, relationships, and conjectures about optimal play strategies and their effects on these parameters.
Contribution
It provides a lower bound for the game domination number on trees, constructs trees with specific domination number properties, and explores the relationship between graphs and their spanning subgraphs.
Findings
Lower bound for game domination number in trees based on order and maximum degree
Existence of trees with specific game domination number pairs
Construction of graphs with spanning trees having significantly different game domination numbers
Abstract
The domination game is played on a graph G. Vertices are chosen, one at a time, by two players Dominator and Staller. Each chosen vertex must enlarge the set of vertices of G dominated to that point in the game. Both players use an optimal strategy---Dominator plays so as to end the game as quickly as possible while Staller plays in such a way that the game lasts as many steps as possible. The game domination number of G is the number of vertices chosen when Dominator starts the game and the Staller-start game domination number of G when Staller starts the game. In this paper these two games are studied when played on trees and spanning subgraphs. A lower bound for the game domination number of a tree in terms of the order and maximum degree is proved and shown to be asymptotically tight. It is shown that for every k, there is a tree T with game domination number k and Staller-start…
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