Bound for the 2-Page Fixed Linear Crossing Number of Hypercube Graph via SDP Relaxation

Bound for the 2-Page Fixed Linear Crossing Number of Hypercube Graph via SDP Relaxation

The crossing number of graph G is the minimum number of edges crossing in any drawing of G in a plane. In this paper we describe a method of finding the bound of 2-page fixed linear crossing number of G. We consider a conflict graph G′ of G. Then, instead of minimizing the crossing number of G, we s...

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Main Authors: A. Suebsriwichai, T. Mouktonglang
Format: Article
Language:English
Published: Hindawi Limited 2017-01-01
Series:Journal of Applied Mathematics
Online Access:http://dx.doi.org/10.1155/2017/7640347
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spelling doaj-823ede38fa324cecbdae0d1803ca90fd2020-11-24T21:11:14ZengHindawi LimitedJournal of Applied Mathematics1110-757X1687-00422017-01-01201710.1155/2017/76403477640347Bound for the 2-Page Fixed Linear Crossing Number of Hypercube Graph via SDP RelaxationA. Suebsriwichai0T. Mouktonglang1Department of Mathematics, Faculty of Science, Chiang Mai University, Chiang Mai 50200, ThailandDepartment of Mathematics, Faculty of Science, Chiang Mai University, Chiang Mai 50200, ThailandThe crossing number of graph G is the minimum number of edges crossing in any drawing of G in a plane. In this paper we describe a method of finding the bound of 2-page fixed linear crossing number of G. We consider a conflict graph G′ of G. Then, instead of minimizing the crossing number of G, we show that it is equivalent to maximize the weight of a cut of G′. We formulate the original problem into the MAXCUT problem. We consider a semidefinite relaxation of the MAXCUT problem. An example of a case where G is hypercube is explicitly shown to obtain an upper bound. The numerical results confirm the effectiveness of the approximation.http://dx.doi.org/10.1155/2017/7640347
collection DOAJ
language English
format Article
sources DOAJ
author A. Suebsriwichai
T. Mouktonglang
spellingShingle A. Suebsriwichai
T. Mouktonglang
Bound for the 2-Page Fixed Linear Crossing Number of Hypercube Graph via SDP Relaxation
Journal of Applied Mathematics
author_facet A. Suebsriwichai
T. Mouktonglang
author_sort A. Suebsriwichai
title Bound for the 2-Page Fixed Linear Crossing Number of Hypercube Graph via SDP Relaxation
title_short Bound for the 2-Page Fixed Linear Crossing Number of Hypercube Graph via SDP Relaxation
title_full Bound for the 2-Page Fixed Linear Crossing Number of Hypercube Graph via SDP Relaxation
title_fullStr Bound for the 2-Page Fixed Linear Crossing Number of Hypercube Graph via SDP Relaxation
title_full_unstemmed Bound for the 2-Page Fixed Linear Crossing Number of Hypercube Graph via SDP Relaxation
title_sort bound for the 2-page fixed linear crossing number of hypercube graph via sdp relaxation
publisher Hindawi Limited
series Journal of Applied Mathematics
issn 1110-757X
1687-0042
publishDate 2017-01-01
description The crossing number of graph G is the minimum number of edges crossing in any drawing of G in a plane. In this paper we describe a method of finding the bound of 2-page fixed linear crossing number of G. We consider a conflict graph G′ of G. Then, instead of minimizing the crossing number of G, we show that it is equivalent to maximize the weight of a cut of G′. We formulate the original problem into the MAXCUT problem. We consider a semidefinite relaxation of the MAXCUT problem. An example of a case where G is hypercube is explicitly shown to obtain an upper bound. The numerical results confirm the effectiveness of the approximation.
url http://dx.doi.org/10.1155/2017/7640347
work_keys_str_mv AT asuebsriwichai boundforthe2pagefixedlinearcrossingnumberofhypercubegraphviasdprelaxation
AT tmouktonglang boundforthe2pagefixedlinearcrossingnumberofhypercubegraphviasdprelaxation
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