A common variable minimax theorem for graphs

TL;DR

This paper investigates the existence and computation of functions that are simultaneously smooth across multiple graphs sharing the same vertices but differing in edges, extending classical minimax theorems to this graph setting.

Contribution

It introduces a common variable minimax theorem for multiple graphs, providing conditions and methods to find functions smooth with respect to all graphs simultaneously.

Findings

Established a minimax theorem for multiple graphs

Provided conditions for the existence of common smooth functions

Suggested algorithms for finding such functions

Abstract

Let $\mathcal{G} = \{G_1 = (V, E_1), \dots, G_m = (V, E_m)\}$ be a collection of $m$ graphs defined on a common set of vertices $V$ but with different edge sets $E_1, \dots, E_m$. Informally, a function $f :V \rightarrow \mathbb{R}$ is smooth with respect to $G_k = (V,E_k)$ if $f(u) \sim f(v)$ whenever $(u, v) \in E_k$. We study the problem of understanding whether there exists a nonconstant function that is smooth with respect to all graphs in $\mathcal{G}$, simultaneously, and how to find it if it exists.