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.

翻译：一对图⟨G₀, G₁⟩允许互证邻近绘图⟨Γ₀, Γ₁⟩当且仅当：(i) Γ_i表示G_i，且(ii) Γ_i中存在边(u,v)当且仅当Γ_{1-i}中没有顶点w与u和v均“过于接近”（i=0,1）。本文通过采用β≥[1,∞]范围内的β-邻近规则，考虑无限多种接近定义，并研究允许互证β-邻近绘图的一对同构树。具体而言，我们证明：任意两棵同构树对任意β∈[1,∞]均允许互证β-邻近绘图。该构造方法具有“鲁棒性”：对某些树对，我们可适当修剪其中一棵树的线性多个叶子，仍保持其互证β-邻近可绘性。值得注意的是，在同构毛虫树且β=1的特殊情形下，我们构造了线性可分离的互证Gabriel绘图。