Minimal locked trees

Brad Ballinger, David Charlton, Erik D. Demaine, Martin L. Demaine, John Iacono, Ching Hao Liu, Sheung Hung Poon

    Research output: Chapter in Book/Report/Conference proceedingConference contribution

    Abstract

    Locked tree linkages have been known to exist in the plane since 1998, but it is still open whether they have a polynomial-time characterization. This paper examines the properties needed for planar trees to lock, with a focus on finding the smallest locked trees according to different measures of complexity, and suggests some new avenues of research for the problem of algorithmic characterization. First we present a locked linear tree with only eight edges. In contrast, the smallest previous locked tree has 15 edges. We further show minimality by proving that every locked linear tree has at least eight edges. We also show that a six-edge tree can interlock with a four-edge chain, which is the first locking result for individually unlocked trees. Next we present several new examples of locked trees with varying minimality results. Finally, we provide counterexamples to two conjectures of [12], [13] by showing the existence of two new types of locked tree: a locked orthogonal tree (all edges horizontal and vertical) and a locked equilateral tree (all edges unit length).

    Original languageEnglish (US)
    Title of host publicationAlgorithms and Data Structures - 11th International Symposium, WADS 2009, Proceedings
    Pages61-73
    Number of pages13
    DOIs
    StatePublished - Sep 14 2009
    Event11th International Symposium on Algorithms and Data Structures, WADS 2009 - Banff, AB, Canada
    Duration: Aug 21 2009Aug 23 2009

    Publication series

    NameLecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
    Volume5664 LNCS
    ISSN (Print)0302-9743
    ISSN (Electronic)1611-3349

    Other

    Other11th International Symposium on Algorithms and Data Structures, WADS 2009
    CountryCanada
    CityBanff, AB
    Period8/21/098/23/09

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    ASJC Scopus subject areas

    • Theoretical Computer Science
    • Computer Science(all)

    Cite this

    Ballinger, B., Charlton, D., Demaine, E. D., Demaine, M. L., Iacono, J., Liu, C. H., & Poon, S. H. (2009). Minimal locked trees. In Algorithms and Data Structures - 11th International Symposium, WADS 2009, Proceedings (pp. 61-73). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 5664 LNCS). https://doi.org/10.1007/978-3-642-03367-4_6