On planar path transformation

doi:10.1016/j.ipl.2007.05.009

Information Processing Letters

Volume 104, Issue 2, 16 October 2007, Pages 59-64

https://doi.org/10.1016/j.ipl.2007.05.009 Get rights and content

Abstract

A flip or edge-replacement is considered as a transformation by which one edge e of a geometric object is removed and an edge f ( $f \neq e$ ) is inserted such that the resulting object belongs to the same class as the original object. Here, we consider Hamiltonian planar paths as geometric objects. A technique is presented for transforming a given planar path into another one for a set S of n points in convex position in the plane. Under these conditions, we show that any planar path can be transformed into another planar path by at most $2 n - 5$ flips. For the case when the points are in general position we provide experimental results regarding transformability of any planar path into another. We show that for $n ⩽ 8$ points in general position any two paths can be transformed into each other. For n points in convex position we show that there are $n 2^{n - 2}$ directed Hamiltonian planar paths. An algorithm is presented which uses flips of size 1 and flips of size 2 to generate all such paths with $O (n)$ time between the generation of two successive paths.

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On planar path transformation

Abstract

Discrete Applied Mathematics

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Computational Geometry: Theory and Applications

Theoretical Computer Science

Discrete Mathematics

Discrete Mathematics

Computational Geometry

Computational Geometry: Theory and Applications