Digraph Placement Games

TL;DR

The paper defines Digraph Placement, a partisan placement game on directed graphs with a blue/Left and red/Right coloring, where moves delete a vertex and its out-neighborhood. It establishes Digraph Placement as a universal ruleset by showing every short partisan game X has a corresponding Digraph Placement G with G = X, using a constructive gadget-based argument that reduces conflict placement games to Digraph placement. As a corollary, winner determination is PSPACE-hard via reductions from Poset Games, and the authors investigate bounds on the size of the smallest graph realizing a given game value, introducing F(b) and related quantities. They further develop upper and lower bounds for these extremal quantities, compute exact values for small birthdays, and outline several open directions for future work, including improving bounds and understanding graph-parameter effects on outcomes.

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 $G = X$.

Figures (3)

• Figure 1: Two Digraph placement games equal to $2 + \mathord\downarrow + \mathord{\ast}$. Here $2 + \mathord\downarrow + \mathord{\ast}$ is chosen because it is an examples of simple, yet non-trivial combinatorial game. In this, and all other figures in this paper, an undirected edge $(u,v)$ implies that there exists directed edges $(u,v)$ and $(v,u)$. Blue vertices are drawn as circles and red vertices are drawn as squares.
• Figure 2: A game of Hackenbush, with literal form $\{ 1-1,-\frac{1}{2} \mathrel\vert \{ 0,1 \mathrel\vert \} \}$, that is not a conflict placement game. This game is not a conflict placement game because the top edge cannot be cut if both bottom edges are cut, but the top edge can be cut if at most one of the bottom edges is cut.
• Figure 3: The Digraph placement game $1\langle 0,H \mathrel\vert 0,H\rangle = \mathord{\ast}2$.

Theorems & Definitions (26)

• Theorem 3.1
• proof
• Corollary 3.2
• proof
• Lemma 4.1: Chapter ii Theorem 3.7 siegel2013combinatorial
• Lemma 4.2: Chapter ii Theorem 3.21 siegel2013combinatorial
• Lemma 4.3
• proof
• Lemma 4.4
• proof