Mathematica Pannonica
Volume/Issue:
Volume 29_NS3: Issue 1
Splitting Edge Partitions of Graphs
Authors:
Balázs Király Institute of Mathematics and Computer Science, Faculty of Natural Sciences, University of Pécs, If júság u. 6, 7624 Pécs, Hungary

Search for other papers by Balázs Király in
Current site
Google Scholar
PubMed Close
and
Sándor Szabó Institute of Mathematics and Computer Science, Faculty of Natural Sciences, University of Pécs, If júság u. 6, 7624 Pécs, Hungary

Search for other papers by Sándor Szabó in
Current site
Google Scholar
PubMed Close
Pages:
37–48
Online Publication Date:
06 Feb 2023
Publication Date:
19 May 2023
Article Category:
Research Article
DOI:
https://doi.org/10.1556/314.2023.00002
Keywords:
Clique number; maximum clique; branching rule; coloring the nodes; preconditioning
Open access

In a typical maximum clique search algorithm when optimality testing is inconclusive a forking takes place. The instance is divided into smaller ones. This is the branching step of the procedure. In order to ensure a balanced work load for the processors for parallel algorithms it is essential that the resulting smaller problems are do not overly vary in difficulty. The so-called splitting partitions of the nodes of the given graph were introduced earlier to meliorate this problem. The paper proposes a splitting partition of the edges for the same purpose. In the lack of available theoretical tools we assess the practical feasibility of constructing suboptimal splitting edge partitions by carrying out numerical experiments. While working with splitting partitions we have realized that they can be utilized as preconditioning tools preliminary to a large scale clique search. The paper will discuss this new found role of the splitting edge partitions as well.

  • [1]

    CARRAGHAN, R., AND PARDALOS, P. M. An exact algorithm for the maximum clique problem. Operation Research Letters 9 (1990), 375–382.

  • [2]

    GAREY, M. R., AND JOHNSON, D. S. Computers and Intractability: A Guide to the Theory of NP-completeness. Freeman, New York, 2003.

  • [3]

    HASSELBERG, J., PARDALOS, P. M., AND VAIRAKTARAKIS, G. Test case generators and compu-tational results for the maximum clique problem. Journal of Global Optimization 3 (1993), 463–482.

    • Search Google Scholar
    • Export Citation
  • [4]

    MATULA, D. W., MARBLE, G., AND ISAACSON, J. D. Graph colouring algorithms. In Graph Theory and Computing, R. C. Read, Ed. Academic Press, New York, 1972.

    • Search Google Scholar
    • Export Citation
  • [5]

    ÖSTERGÅRD, P. R. J. A fast algorithm for the maximum clique problem. Discrete Applied Mathematics 120 (2002), 197–207.

  • [6]

    ÖSTERGÅRD, P. R. J., AND PÖLLÄNEN, A. New results on tripod packings. Discrete and Computa-tional Geometry 61 (2019), 271–284.

  • [7]

    PAPADIMITRIOU, C. H. Computational Complexity. Addison-Wesley Publishing Company, Inc., Reading, MA, 1994.

  • [8]

    SLOANE, N. J. A. Challenge problems: Independent sets in graphs. https://oeis.org/A265032/a265032.html, 2000. Accessed: 2022-10-18.

  • [9]

    SZABÓ, S. Parallel algorithms for finding cliques in a graph. Journal of Physics, Conference Series 268 (2011), 012–030.

  • [10]

    SZABÓ, S. Monoton matrices and finding cliques in a graph. Annales Univ. Sci. Budapest., Sect. Computatorica 41 (2013), 307–322.

  • [11]

    SZABÓ, S. Metric space method for constructing splitting partitions of graphs. Acta Univ. Sapientiae, Informatica 11 (2019), 131–141.

    • Search Google Scholar
    • Export Citation
  • [12]

    SZABÓ, S., AND ZAVÁLNIJ, B. Greedy algorithms for triangle free coloring. AKCE International Journal of Graphs and Combinatorics 9, 2 (2012), 169–186.

    • Search Google Scholar
    • Export Citation
  • [13]

    SZABÓ, S., AND ZAVÁLNIJ, B. Reducing graph coloring to clique search. Asia Pacific Journal of Mathematics 3 (2016), 64–85.

  • [14]

    TOMITA, E., AND SEKI, T. An efficient branch-and-bound algorithm for finding a maximum clique. pp. 278–289.

  • [15]

    WEISSTEIN, E. W. Monotonic matrix. http://mathworld.wolfram.com/MonotonicMatrix.html. In: MathWorld–A Wolfram Web Resource. Accessed: 2022-10-18.

    • Search Google Scholar
    • Export Citation
  • Collapse
  • Expand
Cover Mathematica Pannonica
Mathematica Pannonica
Print ISSN:
0865-2090
Online ISSN:
2786-0752

Editor in Chief: László TÓTH, University of Pécs, Pécs, Hungary

Honorary Editors in Chief:

  • János PINTZ, HUN-REN Alfréd Rényi Institute of Mathematics, Budapest, Hungary
  • † Ferenc SCHIPP, Eötvös Loránd University, Budapest, Hungary and University of Pécs, Pécs, Hungary
  • Sándor SZABÓ, University of Pécs, Pécs, Hungary
     

Deputy Editors in Chief:

  • Erhard AICHINGER, JKU Linz, Linz, Austria
  • Ferenc HARTUNG, University of Pannonia, Veszprém, Hungary
  • Ferenc WEISZ, Eötvös Loránd University, Budapest, Hungary

Editorial Board

  • Attila BÉRCZES, University of Debrecen, Debrecen, Hungary
  • István BERKES, Rényi Institute of Mathematics, Budapest, Hungary
  • Károly BEZDEK, University of Calgary, Calgary, Canada
  • György DÓSA, University of Pannonia, Veszprém, Hungary
  • Balázs KIRÁLY – Managing Editor, University of Pécs, Pécs, Hungary
  • Vedran KRCADINAC, University of Zagreb, Zagreb, Croatia 
  • Željka MILIN ŠIPUŠ, University of Zagreb, Zagreb, Croatia
  • Gábor NYUL, University of Debrecen, Debrecen, Hungary
  • Margit PAP, University of Pécs, Pécs, Hungary
  • István PINK, University of Debrecen, Debrecen, Hungary
  • Mihály PITUK, University of Pannonia, Veszprém, Hungary
  • Lukas SPIEGELHOFER, Montanuniversität Leoben, Leoben, Austria
  • Andrea ŠVOB, University of Rijeka, Rijeka, Croatia
  • Csaba SZÁNTÓ, Babeş-Bolyai University, Cluj-Napoca, Romania
  • Jörg THUSWALDNER, Montanuniversität Leoben, Leoben, Austria
  • Zsolt TUZA, University of Pannonia, Veszprém, Hungary

Advisory Board

  • Szilárd RÉVÉSZ – Chair, HUN-REN Alfréd Rényi Institute of Mathematics, Budapest, Hungary
  • Gabriella BÖHM
  • György GÁT, University of Debrecen, Debrecen, Hungary

University of Pécs,
Faculty of Sciences,
Institute of Mathematics and Informatics
Department of Mathematics
7624 Pécs, Ifjúság útja 6., HUNGARY
(36) 72-503-600 / 4179
ltoth@gamma.ttk.pte.hu

  • Mathematical Reviews
  • Zentralblatt
  • DOAJ

Publication Model Gold Open Access
Online only
Submission Fee none
Article Processing Charge 0 EUR/article (temporarily)
Subscription Information Gold Open Access

Mathematica Pannonica
Language English
Size A4
Year of
Foundation		 1990
Volumes
per Year		 1
Issues
per Year		 2
Founder Akadémiai Kiadó
Founder's
Address		 H-1117 Budapest, Hungary 1516 Budapest, PO Box 245.
Publisher Akadémiai Kiadó
Publisher's
Address		 H-1117 Budapest, Hungary 1516 Budapest, PO Box 245.
Responsible
Publisher		 Chief Executive Officer, Akadémiai Kiadó
ISSN 2786-0752 (Online)
ISSN 0865-2090 (Print)

Monthly Content Usage

Abstract Views Full Text Views PDF Downloads
Nov 2024 0 57 7
Dec 2024 0 26 2
Jan 2025 0 82 3
Feb 2025 0 85 2
Mar 2025 0 58 1
Apr 2025 0 11 1
May 2025 0 0 0