Article in volume
Authors:
Title:
Open locating-dominating sets in circulant graphs
PDFSource:
Discussiones Mathematicae Graph Theory 42(1) (2022) 47-62
Received: 2017-11-11 , Revised: 2019-02-20 , Accepted: 2019-06-21 , Available online: 2019-07-22 , https://doi.org/10.7151/dmgt.2235
Abstract:
Location detection problems have been studied for a variety of applications
including finding faults in multiprocessors, contaminants in public utilities,
intruders in buildings and facilities, and for environmental monitoring using
wireless sensor networks. In each of these applications, the system or
structure can be modeled as a graph, and sensors placed strategically at a
subset of vertices can locate and detect anomalies in the system.
An open locating-dominating set (OLD-set) is a subset of vertices in a graph
in which every vertex in the graph has a non-empty and unique set of neighbors
in the subset. Sensors placed at OLD-set vertices can uniquely detect and locate
disturbances in a system. These sensors can be expensive and, as a result,
minimizing the size of the OLD-set is critical. Circulant graphs, a group of
regular cyclic graphs, are often used to model parallel networks. We prove the
optimal OLD-set size for a particular circulant graph using Hall's Theorem.
We also consider the mixed-weight OLD-set introduced in [R.M. Givens, R.K. Kincaid, W. Mao and G. Yu,
Mixed-weight open locating-dominating sets, in: 2017 Annual Conference
on Information Science and Systems, (IEEE, Baltimor, 2017) 1–6] which
models a system with sensors of varying strengths. To model these systems,
we place weights on the vertices in the graph, representing the strength of
a sensor placed at the corresponding location in the system. We study
particular mixed-weight OLD-sets in cycles, which behave similarly to OLD-sets in
circulant graphs, and show the optimal mixed-weight OLD-set size using the
discharging method.
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Keywords:
open locating-dominating sets, circulant graphs, Hall's Matching Theorem, mixed-weight open locating-dominating sets
References:
- I.F. Akyildiz, D. Pompili and T. Melodia, Underwater acoustic sensor networks: research challenges, Ad Hoc Networks 3 (2005) 257–279.
https://doi.org/10.1016/j.adhoc.2005.01.004 - K. Appel and W. Haken, Every planar map is four colorable, Part I: Discharging, Illinois J. Math. 21 (1977) 249–490.
https://doi.org/10.1215/ijm/1256049011 - T.Y. Berger-Wolf, W.E. Hart and J. Saia, Discrete sensor placement problems in distribution networks, Math. Comput. Modelling 42 (2005) 1385-1396.
https://doi.org/10.1016/j.mcm.2005.03.005 - N. Bertrand, I. Charon, O. Hudry and A. Lobstein, Identifying and locating-dominating codes on chains and cycles, European J. Combin. 25 (2004) 969–987.
https://doi.org/10.1016/j.ejc.2003.12.013 - D. Cranston and G. Yu, A new lower bound on the density of vertex identifying codes for the infinite hexagonal grid, Electron. J. Combin. 16 (2009) $#$R113.
- A. Cukierman and G. Yu, New bounds on the minimum density of an identifying code for the infinite hexagonal grid, Discrete Appl. Math. 161 (2013) 2910–2924.
https://doi.org/10.1016/j.dam.2013.06.002 - D. Di Palma, L. Bencini, G. Collodi, G. Manes, F. Chiti, {R. Fantacci and A. Manes}, Disctributed monitoring systems for agriculture based on wireless sensor network technology, Internat. J. Advances in Networks and Services 3 (2010) 18–28.
- F. Foucaud and M.A. Henning, Location-domination and matching in cubic graphs, Discrete Math. 339 (2016) 1221–1231.
https://doi.org/10.1016/j.disc.2015.11.016 - R.M. Givens, R.K. Kincaid, W. Mao and G. Yu, Mixed-weight open locating-dominating sets, in: 2017 Annual Conference on Information Science and Systems, (IEEE, Baltimor 2017) 1–6.
https://doi.org/10.1109/CISS.2017.7926110 - P. Hall, On representatives of subsets, J. London Math. Soc. (2) 10 (1935) 26–30.
https://doi.org/10.1112/jlms/s1-10.37.26 - I. Honkala, T. Laihonen and S. Ranto, On locating-dominating codoes in binary Hamming spaces, Discrete Math. Theor. Comput. Sci. 6 (2004) 265–282.
- S. Janson and T. Laihonen, On the size of identifying codes in binary hypercubes, J. Combin. Theory Ser. A 116 (2009) 1087–1096.
https://doi.org/10.1016/j.jcta.2009.02.004 - R.K. Kincaid, A. Oldham and G. Yu, Optimal open-locating-dominating sets in infinite triangular grids, Discrete Appl. Math. 193 (2015) 139–144.
https://doi.org/10.1016/j.dam.2015.04.024 - M. Laifenfeld and A. Trachtenberg, Identifying codes and covering problems, IEEE Trans. Inform. Theory 54 (2008) 3929–3950.
https://doi.org/10.1109/TIT.2008.928263 - M. Laifenfeld, A. Trachtenberg, R. Cohen and D. Starobinski, Joint monitoring and routing in wireless sensor networks using robust identifying codes, in: 2007 4th International Conference on Broadband Communications, Networks and System, (IEEE, Releigh 2007) 197–206.
https://doi.org/10.1109/BROADNETS.2007.4550425 - A. Lobstein, Watching systems, identyfying, locating-dominating and discriminating codes in graphs (a bibliography), 2017.
- A. Mainwaring, D. Culler, J. Polastre, R. Szewczyk and J. Anderson, Wireless sensor networks for habitat monitoring, in: Proc. First ACM International Workshop on Wireless Sensor Networks and Application, (ACM, Atlanta 2002) 88–97.
https://doi.org/10.1145/570738.570751 - P.D. Manual, Locating and liar domination of circulant networks, Ars Combin. 101 (2011) 309–320.
- P. Padhy, K. Martinez, A. Riddoch, H.L.R. Ong and J.K. Hart, Glacial environment monitoring using sensor networks, in: Proc. Real-World Wireless Sensor Networks, (ACM, Stockholm 2005) 10–14.
- S. Ray, D. Starobinski, A. Trachtenberg and R. Ungrangsi, Robust location detection with sensor networks, IEEE J. Selected Areas in Communications 22 (2004) 1016–1025.
https://doi.org/10.1109/JSAC.2004.830895 - S.J. Seo and P.J. Slater, Open neighborhood locating-dominating sets, Australas. J. Combin. 46 (2010) 109–120.
- S.J. Seo and P.J. Slater, Open neighborhood locating-domination for grid-like graphs, Bull. Inst. Combin. Appl. 65 (2012) 89–100.
- E. Vallejo, R. Beivide and C. Martinez, Practicable layouts for optimal circulant graphs, in: 13th Euromicro Conference on Parallel, Distributed and Network-Based Processing, (IEEE, Lugano 2005) 118–125.
https://doi.org/10.1109/EMPDP.2005.32
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