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⟩ 允许互见证邻近绘制 ⟨Γ₀, Γ₁⟩，当且仅当：(i) Γᵢ 表示 Gᵢ，且 (ii) Γᵢ 中存在边 (u,v) 当且仅当 Γ₁ᵢ 中没有顶点 w 同时与 u 和 v“过于接近”（i=0,1）。本文采用 β-邻近规则，对任意 β ∈ [1,∞] 考虑无限多种接近性定义，并研究允许互见证 β-邻近绘制的同构树对。具体而言，我们证明任意两棵同构树在任意 β ∈ [1,∞] 下均允许互见证 β-邻近绘制。该构造方法具有“鲁棒性”：对于某些树对，我们可以从其中一棵树中适当剪除线性数量的叶子，仍保持其互见证 β-邻近可绘制性。值得关注的是，在同构毛虫树且 β=1 的特殊情况下，我们构造了线性可分离的互见证 Gabriel 绘制。