Mutual Witness Proximity Drawings of Isomorphic Trees
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A pair ⟨ G_0, G_1 ⟩ of graphs admits a mutual witness proximity drawing ⟨Γ_0, Γ_1 ⟩ 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∈ [1,∞] 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∈ [1,∞]. 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.
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