Notes: Packing cycles through prescribed vertices

Authors:
Naonori Kakimura;Ken-ichi Kawarabayashi;Dániel Marx
Affiliations:
Department of Mathematical Informatics, Graduate School of Information Science and Technology, University of Tokyo, 7-3-1, Hongo, Bunkyo-ku, Tokyo 113-8656, Japan;National Institute of Informatics, Japan;Institut für Informatik, Humboldt-Universität zu Berlin, Germany
Venue:
Journal of Combinatorial Theory Series B
Year:
2011

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Abstract

The well-known theorem of Erdos and Posa says that a graph G has either k vertex-disjoint cycles or a vertex set X of order at most f(k) such that G@?X is a forest. Starting with this result, there are many results concerning packing and covering cycles in graph theory and combinatorial optimization. In this paper, we generalize Erdos-Posa@?s result to cycles that are required to go through a set S of vertices. Given an integer k and a vertex subset S (possibly unbounded number of vertices) in a given graph G, we prove that either G has k vertex-disjoint cycles, each of which contains at least one vertex of S, or G has a vertex set X of order at most f(k)=40k^2log"2k such that G@?X has no cycle that intersects S.