# Adaptive Majority Problems for Restricted Query Graphs and for Weighted Sets

## TL;DR

This paper investigates the minimum number of edge queries needed to find a majority vertex in graphs, extending previous results from complete graphs to general graphs, with specific bounds for trees and other graph classes.

## Contribution

It generalizes the majority problem to arbitrary graphs, establishing bounds for trees and constructing graphs with optimal query complexity and sparse structure.

## Key findings

- For even trees, the query complexity equals the number of vertices minus one.
- For odd trees, the complexity is between n-65 and n-2.
- Constructed graphs with optimal complexity and O(nb(n)) edges.

## Abstract

Suppose that the vertices of a graph $G$ are colored with two colors in an unknown way. The color that occurs on more than half of the vertices is called the majority color (if it exists), and any vertex of this color is called a majority vertex. We study the problem of finding a majority vertex (or show that none exists) if we can query edges to learn whether their endpoints have the same or different colors. Denote the least number of queries needed in the worst case by $m(G)$. It was shown by Saks and Werman that $m(K_n)=n-b(n)$, where $b(n)$ is the number of 1's in the binary representation of $n$. In this paper, we initiate the study of the problem for general graphs. The obvious bounds for a connected graph $G$ on $n$ vertices are $n-b(n)\le m(G)\le n-1$. We show that for any tree $T$ on an even number of vertices we have $m(T)=n-1$ and that for any tree $T$ on an odd number of vertices, we have $n-65\le m(T)\le n-2$. Our proof uses results about the weighted version of the problem for $K_n$, which may be of independent interest. We also exhibit a sequence $G_n$ of graphs with $m(G_n)=n-b(n)$ such that $G_n$ has $O(nb(n))$ edges and $n$ vertices.