Digraph Placement Games

Abstract

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 $G = X$.