Sign games on graphs

TL;DR

This work introduces the Sign Game on simple undirected graphs, where two players alternately label vertices with $+1$ or $-1$ and the final edge-score $s(G)=\sum_{uv\in E(G)} x_u x_v$ determines the winner ($s(G)>0$ for P, $s(G)<0$ for N, $s(G)=0$ for a draw). It develops SG-equivalence as a reduction technique to replace subgraphs by SG-equivalent gadgets without changing the outcome, enabling tractable analysis on larger graphs. The paper provides complete classifications of outcomes for complete graphs $K_n$, star graphs $S_n$, complete bipartite graphs $K_{m,n}$, path graphs $P_n$, and cycle graphs $C_n$, using mirroring strategies and segment-based decompositions (notably for cycles via 4-segments). Overall, the results reveal parity- and structure-dependent strategies, with notable findings such as N's dominance on most complete graphs, parity-driven outcomes on stars and paths, and a four-case mod $4$ rule for cycles, illustrating rich strategic structure and suggesting broad avenues for generalization and extensions.

Abstract

We define the Sign Game as a two-player game played on a simple undirected mathematical graph $G$. The players alternate turns, assigning vertices of $G$ either $1$ or $-1$, and edges take on the value of the product of their endvertices. The game ends when all vertices are assigned values, and the score of the game is the sum of all edge values. One player's goal is to make the score positive while the other's is to make the score negative. In this paper we investigate the game being played on various types of graphs, determining outcomes and winning strategies.

Figures (7)

• Figure 1.1: Player N wins
• Figure 1.2: SG-equivalent graphs $G_1$ and $G_2$
• Figure 2.1: The graph $P_4$
• Figure 2.2: SG-equivalent graphs: path and union of paths
• Figure 2.3: SG-equivalent graphs: $C_4$ and $P_5$

Theorems & Definitions (14)

• Lemma 2.1
• Theorem 2.2
• proof
• Theorem 2.3
• proof
• Theorem 2.4
• Lemma 2.5
• Theorem 2.6
• proof
• Theorem 2.7