TY - JOUR

T1 - Reconstruction of the path graph

AU - Keller, Chaya

AU - Stein, Yael

N1 - Publisher Copyright:
© 2018 Elsevier B.V.

PY - 2018/6

Y1 - 2018/6

N2 - Let P be a set of n≥5 points in convex position in the plane. The path graph G(P) of P is an abstract graph whose vertices are non-crossing spanning paths of P, such that two paths are adjacent if one can be obtained from the other by deleting an edge and adding another edge. We prove that the automorphism group of G(P) is isomorphic to Dn, the dihedral group of order 2n. The heart of the proof is an algorithm that first identifies the vertices of G(P) that correspond to boundary paths of P, where the identification is unique up to an automorphism of K(P) as a geometric graph, and then identifies (uniquely) all edges of each path represented by a vertex of G(P). The complexity of the algorithm is O(NlogN) where N is the number of vertices of G(P).

AB - Let P be a set of n≥5 points in convex position in the plane. The path graph G(P) of P is an abstract graph whose vertices are non-crossing spanning paths of P, such that two paths are adjacent if one can be obtained from the other by deleting an edge and adding another edge. We prove that the automorphism group of G(P) is isomorphic to Dn, the dihedral group of order 2n. The heart of the proof is an algorithm that first identifies the vertices of G(P) that correspond to boundary paths of P, where the identification is unique up to an automorphism of K(P) as a geometric graph, and then identifies (uniquely) all edges of each path represented by a vertex of G(P). The complexity of the algorithm is O(NlogN) where N is the number of vertices of G(P).

KW - Convex geometric graphs

KW - Geometric graphs

KW - Hamiltonian paths

KW - Reconstruction

UR - http://www.scopus.com/inward/record.url?scp=85042561838&partnerID=8YFLogxK

U2 - 10.1016/j.comgeo.2018.02.002

DO - 10.1016/j.comgeo.2018.02.002

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AN - SCOPUS:85042561838

SN - 0925-7721

VL - 72

SP - 1

EP - 10

JO - Computational Geometry: Theory and Applications

JF - Computational Geometry: Theory and Applications

ER -