Total connected domination game

TL;DR

This paper studies the total connected domination game on graphs, establishing bounds on the game domination number, classifying graphs into classes based on these bounds, and characterizing certain classes of graphs.

Contribution

It introduces a classification of graphs based on the total connected domination game number and characterizes classes of trees and bipartite graphs within this framework.

Findings

The total connected game domination number differs from the connected game domination number by at most 2.

All connected Cartesian and direct product graphs with minimum degree at least 2 are Class 0.

No tree belongs to Class 2, and Class 1 trees are characterized.

Abstract

The (total) connected domination game on a graph $G$ is played by two players, Dominator and Staller, according to the standard (total) domination game with the additional requirement that at each stage of the game the selected vertices induce a connected subgraph of $G$. If Dominator starts the game and both players play optimally, then the number of vertices selected during the game is the (total) connected game domination number ($\gamma_{\rm tcg}(G)$) $\gamma_{\rm cg}(G)$ of $G$. We show that $\gamma_{\rm tcg}(G)\in \{\gamma_{\rm cg}(G), \gamma_{\rm cg}(G) + 1, \gamma_{\rm cg}(G) + 2\}$, and consequently define $G$ as Class $i$ if $\gamma_{\rm tcg}(G) = \gamma_{\rm cg} + i$ for $i \in \{0,1,2\}$. A large family of Class $0$ graphs is constructed which contains all connected Cartesian product graphs and connected direct product graphs with minumum degree at least $2$. We show that no tree is Class $2$ and characterize Class $1$ trees. We provide an infinite family of Class $2$ bipartite graphs.