Mutual Witness Proximity Drawings of Isomorphic Trees

TL;DR

This paper proves that all pairs of isomorphic trees can be represented with mutual witness proximity drawings under various closeness rules, including special cases like caterpillars with Gabriel drawings, using a robust, constructive approach.

Contribution

It introduces a comprehensive framework for mutual witness proximity drawings of isomorphic trees for any proximity parameter, including a robust construction method and special case results.

Findings

All isomorphic trees admit mutual witness $eta$-proximity drawings for any $eta$.

A robust pruning technique preserves mutual witness $eta$-proximity drawability.

Existence of linearly separable mutual witness Gabriel drawings for isomorphic caterpillars.

Abstract

A pair $\langle G_0, G_1 \rangle$ of graphs admits a mutual witness proximity drawing $\langle \Gamma_0, \Gamma_1 \rangle$ when: (i) $\Gamma_i$ represents $G_i$, and (ii) there is an edge $(u,v)$ in $\Gamma_i$ if and only if there is no vertex $w$ in $\Gamma_{1-i}$ that is ``too close'' to both $u$ and $v$ ($i=0,1$). In this paper, we consider infinitely many definitions of closeness by adopting the $\beta$-proximity rule for any $\beta \in [1,\infty]$ and study pairs of isomorphic trees that admit a mutual witness $\beta$-proximity drawing. Specifically, we show that every two isomorphic trees admit a mutual witness $\beta$-proximity drawing for any $\beta \in [1,\infty]$. The constructive technique can be made ``robust'': For some tree pairs we can suitably prune linearly many leaves from one of the two trees and still retain their mutual witness $\beta$-proximity drawability. Notably, in the special case of isomorphic caterpillars and $\beta=1$, we construct linearly separable mutual witness Gabriel drawings.