Survey: Cycle bases in graphs characterization, algorithms, complexity, and applications

  • Authors:

  • Telikepalli Kavitha;Christian Liebchen;Kurt Mehlhorn;Dimitrios Michail;Romeo Rizzi;Torsten Ueckerdt;Katharina A. Zweig

  • Affiliations:

  • Department of Computer Science and Automation, Indian Institute of Science, Bangalore, India;DB Schenker Rail Deutschland AG, Mainz, Germany;Max-Planck-Institut für Informatik, Saarbrücken, Germany;Department of Informatics and Telematics, Harokopio University of Athens 89, Harokopou Street 17671, Kallithea, Athens, Greece;Dipartimento di Matematica ed Informatica (DIMI), Universití degli Studi di Udine, Udine, Italy;Technische Universität Berlin, Berlin, Germany;Department of Biological Physics, Eötvös Lorand University, Budapest, Hungary

  • Venue:
  • Computer Science Review
  • Year:
  • 2009

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Abstract

Cycles in graphs play an important role in many applications, e.g., analysis of electrical networks, analysis of chemical and biological pathways, periodic scheduling, and graph drawing. From a mathematical point of view, cycles in graphs have a rich structure. Cycle bases are a compact description of the set of all cycles of a graph. In this paper, we survey the state of knowledge on cycle bases and also derive some new results. We introduce different kinds of cycle bases, characterize them in terms of their cycle matrix, and prove structural results and a priori length bounds. We provide polynomial algorithms for the minimum cycle basis problem for some of the classes and prove APX-hardness for others. We also discuss three applications and show that they require different kinds of cycle bases.