TY - JOUR

T1 - Mutual witness proximity graphs

AU - Aronov, Boris

AU - Dulieu, Muriel

AU - Hurtado, Ferran

N1 - Funding Information:
Boris Aronov and Muriel Dulieu were partially supported by grant No. 2006/194 from the United States – Israel Binational Science Foundation and by NSF Grants CCF-08-30691 , CCF-11-17336 , and CCF-12-18791 . Boris Aronov was also supported by NSA MSP Grant H98230- 10-1-0210 . Ferran Hurtado is partially supported by projects MEC MTM2006-01267 , MICINN MTM2009-07242 , MINECO MTM2012-30951 , Generalitat de Catalunya DGR 2009SGR1040 , and ESF EUROCORES programme EuroGIGA, CRP ComPoSe: MICINN Project EUI-EURC-2011-4306 , for Spain.
Copyright:
Copyright 2014 Elsevier B.V., All rights reserved.

PY - 2014/10

Y1 - 2014/10

N2 - This paper describes one variation on witness proximity graphs called mutual witness proximity graphs. Two witness proximity graphs are said to be mutual when, given two sets of points A and B, A is the vertex set of the first graph and the witness set of the second one, while B is the witness set of the first graph and the vertex set of the second one. We show that in the union of two mutual witness Delaunay graphs, there are always at least n-22 edges, where n=|A|+|B|, which is tight in the worst case. We also show that if two mutual witness Delaunay graphs are complete, then the sets A and B are circularly separable; if two mutual witness Gabriel graphs are complete, then the sets A and B are linearly separable; but two mutual witness rectangle graphs might be complete, with A and B not linearly separable.

AB - This paper describes one variation on witness proximity graphs called mutual witness proximity graphs. Two witness proximity graphs are said to be mutual when, given two sets of points A and B, A is the vertex set of the first graph and the witness set of the second one, while B is the witness set of the first graph and the vertex set of the second one. We show that in the union of two mutual witness Delaunay graphs, there are always at least n-22 edges, where n=|A|+|B|, which is tight in the worst case. We also show that if two mutual witness Delaunay graphs are complete, then the sets A and B are circularly separable; if two mutual witness Gabriel graphs are complete, then the sets A and B are linearly separable; but two mutual witness rectangle graphs might be complete, with A and B not linearly separable.

KW - Computational geometry

KW - Delaunay graphs

KW - Gabriel graphs

KW - Proximity graphs

KW - Rectangle of influence graphs

KW - Witness graphs

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U2 - 10.1016/j.ipl.2014.04.001

DO - 10.1016/j.ipl.2014.04.001

M3 - Article

AN - SCOPUS:84899854890

VL - 114

SP - 519

EP - 523

JO - Information Processing Letters

JF - Information Processing Letters

SN - 0020-0190

IS - 10

ER -