Cycle Bases in Graphs Characterization, Algorithms, Complexity, and Applications
Cycle Bases in Graphs Characterization, Algorithms, Complexity, and Applications Telikepalli Kavitha∗ Christian Liebchen† Kurt Mehlhorn‡ Dimitrios Michail Romeo Rizzi§ Torsten Ueckerdt¶ Katharina A. Zweigk August 25, 2009 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 apriori length bounds. We provide polynomial algorithms for the minimum cycle basis problem for some of the classes and prove -hardness for others. We also discuss three applications and show that they requirAPXe different kinds of cycle bases. Contents 1 Introduction 3 2 Definitions 5 3 Classification of Cycle Bases 10 3.1 Existence ...................................... 10 3.2 Characterizations............................... 11 3.3 SimpleExamples .................................. 15 3.4 VariantsoftheMCBProblem. 17 3.5 Directed and GF (p)-Bases............................. 20 3.6 CircuitsversusCycles . .. .. .. .. .. .. .. 21 3.7 Reductions ..................................... 24 ∗Department of Computer Science and Automation, Indian Institute of Science, Bangalore, India †DB Schenker Rail Deutschland AG, Mainz, Germany. Part of this work was done at the DFG Research Center Matheon in Berlin ‡Max-Planck-Institut f¨ur Informatik, Saarbr¨ucken, Germany §Dipartimento di Matematica ed Informatica (DIMI), Universit degli Studi di Udine, Udine, Italy ¶Technische Universit¨at Berlin, Berlin, Germany, Supported by .
[Show full text]