Abstract
The total domination game is played on a simple graph [Formula: see text] with no isolated vertices by two players, Dominator and Staller, who alternate choosing a vertex in [Formula: see text]. Each chosen vertex totally dominates its neighbors. In this game, each chosen vertex must totally dominates at least one new vertex not totally dominated before. The game ends when all vertices in [Formula: see text] are totally dominated. Dominator’s goal is to finish the game as soon as possible, and Staller’s goal is to prolong it as much as possible. The game total domination number is the number of chosen vertices when both players play optimally, denoted by [Formula: see text] when Dominator starts the game and denoted by [Formula: see text] when Staller starts the game. If a vertex [Formula: see text] in [Formula: see text] is declared to be already totally dominated, then we denote this graph by [Formula: see text]. A total domination game critical graph is a graph [Formula: see text] for which [Formula: see text] holds for every vertex [Formula: see text] in [Formula: see text]. If [Formula: see text], then [Formula: see text] is called [Formula: see text]-[Formula: see text]-critical. In this work, we characterize some 4-[Formula: see text]-critical graphs.
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Disclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.