The Sitting Closer to Friends than Enemies Problem in Trees

TL;DR

This paper investigates the conditions under which a signed graph can be embedded into a real tree so that positive neighbors are closer than negative ones, linking it to intersection representations by unit balls.

Contribution

It establishes a characterization of valid distance drawings in real trees via intersection representations, connecting positive edge subgraphs to geometric ball intersections.

Findings

Complete signed graphs have valid distance drawings iff their positive subgraphs are intersection graphs of unit balls in a real tree.

Intersection representations by unit, proper, and arbitrary balls in a real tree are equivalent.

The work provides a geometric perspective on the Sitting Closer to Friends than Enemies problem in trees.

Abstract

A metric space $\mathcal{T}$ is a \emph{real tree} if for any pair of points $x, y \in \mathcal{T}$ all topological embeddings $\sigma$ of the segment $[0,1]$ into $\mathcal{T}$, such that $\sigma (0)=x$ and $\sigma (1)=y$, have the same image (which is then a geodesic segment from $x$ to $y$). A \emph{signed graph} is a graph where each edge has a positive or negative sign. The \emph{Sitting Closer to Friends than Enemies} problem in trees has a signed graph $S$ as an input. The purpose is to determine if there exists an injective mapping (called \emph{valid distance drawing}) from $V(S)$ to the points of a real tree such that, for every $u \in V(S)$, for every positive neighbor $v$ of $u$, and negative neighbor $w$ of $u$, the distance between $v$ and $u$ is smaller than the distance between $w$ and $u$. In this work, we show that a complete signed graph has a valid distance drawing in a real tree if and only if its subgraph composed of all (and only) its positive edges has an intersection representation by unit balls in a real tree. Besides, as an instrumental result, we show that a graph has an intersection representation by unit balls in a real tree if and only if it has an intersection representation by proper balls, and if and only if it has an intersection representation by arbitrary balls in a real tree.