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Connected domination game: Predomination, Staller-start game, and lexicographic products

The connected domination game was introduced in 2019 by Borowiecki,
 Fiedorowicz and Sidorowicz as another variation of the domination game. We
 answer a problem from their paper regarding the relation between the number
 of moves in a game where Dominator/Staller starts the game. Additionally, we
 study the relation to the diameter and present graphs with small game
 connected domination number. We also determine the value on the
 lexicographic products, and consider the effect of predomination of a
 vertex.