Digraph Placement Games
Fuente:
arXiv
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2024
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866910910090575872 |
|---|---|
| author | Clow, Alexander McKay, Neil A |
| author_facet | Clow, Alexander McKay, Neil A |
| contents | This paper considers a natural ruleset for playing a partisan combinatorial game on a directed graph, which we call Digraph Placement. Given a digraph $G$ with a not necessarily proper $2$-coloring of $V(G)$, the Digraph Placement game played on $G$ by the players Left and Right, who play alternately, is defined as follows. On her turn, Left chooses a blue vertex which is deleted along with all of its out-neighbours. On his turn Right chooses a red vertex, which is deleted along with all of its out-neighbours. A player loses if on their turn they cannot move. We show constructively that Digraph Placement is a universal partisan ruleset; for all partisan combinatorial games $X$ there exists a Digraph Placement game, $G$, such that $G = X$. Digraph Placement and many other games including Nim, Poset Game, Col, Node Kayles, Domineering, and Arc Kayles are instances of a class of placement games that we call conflict placement games. We prove that $X$ is a conflict placement game if and only if it has the same literal form as a Digraph Placement game. A corollary of this is that deciding the winner of a Digraph Placement game is PSPACE-hard. Next, for a game value $X$ we prove bounds on the order of a smallest Digraph Placement game $G$ such that $G = X$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_12219 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Digraph Placement Games Clow, Alexander McKay, Neil A Combinatorics 91A46, 05C57 This paper considers a natural ruleset for playing a partisan combinatorial game on a directed graph, which we call Digraph Placement. Given a digraph $G$ with a not necessarily proper $2$-coloring of $V(G)$, the Digraph Placement game played on $G$ by the players Left and Right, who play alternately, is defined as follows. On her turn, Left chooses a blue vertex which is deleted along with all of its out-neighbours. On his turn Right chooses a red vertex, which is deleted along with all of its out-neighbours. A player loses if on their turn they cannot move. We show constructively that Digraph Placement is a universal partisan ruleset; for all partisan combinatorial games $X$ there exists a Digraph Placement game, $G$, such that $G = X$. Digraph Placement and many other games including Nim, Poset Game, Col, Node Kayles, Domineering, and Arc Kayles are instances of a class of placement games that we call conflict placement games. We prove that $X$ is a conflict placement game if and only if it has the same literal form as a Digraph Placement game. A corollary of this is that deciding the winner of a Digraph Placement game is PSPACE-hard. Next, for a game value $X$ we prove bounds on the order of a smallest Digraph Placement game $G$ such that $G = X$. |
| title | Digraph Placement Games |
| topic | Combinatorics 91A46, 05C57 |
| url | https://arxiv.org/abs/2407.12219 |