Mutual Witness Proximity Drawings of Isomorphic Trees

A pair $\langle G_0, G_1 \rangle$ of graphs admits a mutual witness proximity drawing $\langle Γ_0, Γ_1 \rangle$ when: (i) $Γ_i$ represents $G_i$, and (ii) there is an edge $(u,v)$ in $Γ_i$ if and only if there is no vertex $w$ in $Γ_{1-i}$ 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 $β$-proximity rule for any $β\in [1,\infty]$ and study pairs of isomorphic trees that admit a mutual witness $β$-proximity drawing. Specifically, we show that every two isomorphic trees admit a mutual witness $β$-proximity drawing for any $β\in [1,\infty]$. 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 $β$-proximity drawability. Notably, in the special case of isomorphic caterpillars and $β=1$, we construct linearly separable mutual witness Gabriel drawings.