Mutual Witness Proximity Drawings of Isomorphic Trees
Carolina Haase, Philipp Kindermann, William J. Lenhart and, Giuseppe Liotta

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 of graphs admits a mutual witness proximity drawing when: (i) represents , and (ii) there is an edge in if and only if there is no vertex in that is ``too close'' to both and (). In this paper, we consider infinitely many definitions of closeness by adopting the -proximity rule for any and study pairs of isomorphic trees that admit a mutual witness -proximity drawing. Specifically, we show that every two isomorphic trees admit a mutual witness -proximity drawing for any . 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 -proximity drawability.…
Peer Reviews
No public reviews on file for this paper yet. If you reviewed it on a platform where reviews are public (OpenReview, ICLR, NeurIPS, ICML), you can paste yours below so the community can read it here.
Videos
No videos yet. Explain this paper in a talk, walkthrough, or lecture? Add one.
Taxonomy
TopicsComputational Geometry and Mesh Generation · Advanced Graph Theory Research · Advanced Numerical Analysis Techniques
