Maker-Breaker total domination number
Fuente:
arXiv
Saved in:
| Main Authors: | , , , |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866909702082789376 |
|---|---|
| author | Divakaran, Athira James, Tijo Klavžar, Sandi Nair, Latha S |
| author_facet | Divakaran, Athira James, Tijo Klavžar, Sandi Nair, Latha S |
| contents | The Maker-Breaker total domination number, $γ_{\rm MBT}(G)$, 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, $γ_{\rm MBT}'(G)$, is defined analogously for the game in which Staller starts. Upper and lower bounds on $γ_{\rm MBT}(G)$ and on $γ_{\rm MBT}'(G)$ 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 $γ_{\rm MB}(G)=k$ and $γ_{\rm MBT}(G)=\ell$, (ii) there exists a connected graph $G'$ with $γ_{\rm MB}'(G')=k$ and $γ_{\rm MBT}'(G')=\ell$, and (iii) there there exists a connected graph $G''$ with $γ_{\rm MBT}(G'')=k$ and $γ_{\rm MBT}'(G'')=\ell$. Here, $γ_{\rm MB}$ and $γ_{\rm MB}'$ are corresponding invariants for the Maker-Breaker domination game. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_17341 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Maker-Breaker total domination number Divakaran, Athira James, Tijo Klavžar, Sandi Nair, Latha S Combinatorics The Maker-Breaker total domination number, $γ_{\rm MBT}(G)$, 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, $γ_{\rm MBT}'(G)$, is defined analogously for the game in which Staller starts. Upper and lower bounds on $γ_{\rm MBT}(G)$ and on $γ_{\rm MBT}'(G)$ 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 $γ_{\rm MB}(G)=k$ and $γ_{\rm MBT}(G)=\ell$, (ii) there exists a connected graph $G'$ with $γ_{\rm MB}'(G')=k$ and $γ_{\rm MBT}'(G')=\ell$, and (iii) there there exists a connected graph $G''$ with $γ_{\rm MBT}(G'')=k$ and $γ_{\rm MBT}'(G'')=\ell$. Here, $γ_{\rm MB}$ and $γ_{\rm MB}'$ are corresponding invariants for the Maker-Breaker domination game. |
| title | Maker-Breaker total domination number |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2507.17341 |