The effect of local majority on global majority in connected graphs

TL;DR

This paper investigates how local majority conditions on small subgraphs influence the global majority in infinite families of connected graphs, classifying the forcing behavior of different subgraph sizes across various graph classes.

Contribution

It introduces a classification framework for the influence of local majority on global majority in connected graphs, analyzing forcing, weakly forcing, and collapsing behaviors for different subgraph sizes.

Findings

Classifies forcing behavior for all connected graphs.

Analyzes forcing for graphs with bounded maximum degree.

Studies forcing for graphs with fixed average degree.

Abstract

Let ${\mathcal G}$ be an infinite family of connected graphs and let $k$ be a positive integer. We say that $k$ is ${\it forcing}$ for ${\mathcal G}$ if for all $G \in {\mathcal G}$ but finitely many, the following holds. Any $\{-1,1\}$-weighing of the edges of $G$ for which all connected subgraphs on $k$ edges are positively weighted implies that $G$ is positively weighted. Otherwise, we say that it is ${\it weakly~forcing}$ for ${\mathcal G}$ if any such weighing implies that the weight of $G$ is bounded from below by a constant. Otherwise we say that $k$ ${\it collapses}$ for ${\mathcal G}$. We classify $k$ for some of the most prominent classes of graphs, such as all connected graphs, all connected graphs with a given maximum degree and all connected graphs with a given average degree.