Maker-Breaker total domination number

TL;DR

This paper introduces and analyzes the Maker-Breaker total domination number in graphs, establishing bounds, sharpness, and existence results for various configurations of the game parameters.

Contribution

It defines the Maker-Breaker total domination number and its variants, providing bounds, sharpness proofs, and existence theorems for graphs with specified parameters.

Findings

Bounds on $oldsymbol{oldsymbol{ ext{}}} ext{γ}_{ ext{MBT}}(G)$ and $ ext{γ}_{ ext{MBT}}'(G)$ are established.

Sharpness of bounds is demonstrated through constructions.

Existence of graphs with prescribed domination parameters is proved.

Abstract

The Maker-Breaker total domination number, $\gamma_{\rm MBT}(G)$, of a graph $G$ is introduced as the minimum number of moves of Dominator to win the Maker-Breaker total domination game, provided that he has a winning strategy and is the first to play. The Staller-start Maker-Breaker total domination number, $\gamma_{\rm MBT}'(G)$, is defined analogously for the game in which Staller starts. Upper and lower bounds on $\gamma_{\rm MBT}(G)$ and on $\gamma_{\rm MBT}'(G)$ are provided and demonstrated to be sharp. It is proved that for any pair of integers $(k,\ell)$ with $2\leq k\leq \ell$, (i) there exists a connected graph $G$ with $\gamma_{\rm MB}(G)=k$ and $\gamma_{\rm MBT}(G)=\ell$, (ii) there exists a connected graph $G'$ with $\gamma_{\rm MB}'(G')=k$ and $\gamma_{\rm MBT}'(G')=\ell$, and (iii) there there exists a connected graph $G''$ with $\gamma_{\rm MBT}(G'')=k$ and $\gamma_{\rm MBT}'(G'')=\ell$. Here, $\gamma_{\rm MB}$ and $\gamma_{\rm MB}'$ are corresponding invariants for the Maker-Breaker domination game.