# Connected domination game: predomination, Staller-start game, and lexicographic products

## TL;DR

This paper studies the connected domination game, analyzing the impact of starting player, graph diameter, predomination, and lexicographic products on the game domination number, providing new bounds and exact values.

## Contribution

It resolves an open problem about the relation between starting players and game length, and determines the connected domination number for lexicographic product graphs.

## Key findings

- Relation between Dominator and Staller starting moves clarified.
- Graphs with small connected game domination number identified.
- Exact values for lexicographic product graphs established.

## Abstract

The connected domination game was recently introduced by Borowiecki, Fiedorowicz and Sidorowicz as another variation of the domination game. The rules are essentially the same, except that the set of played vertices must be connected at all stages of the 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. In this paper we also study the relation to the diameter and present graphs with small connected game domination number. We determine the values on the lexicographic product graphs, and consider the effect of predomination of a vertex on the connected game domination number.