TY - JOUR

T1 - Packing cycles through prescribed vertices under modularity constraints

AU - Kakimura, Naonori

AU - Kawarabayashi, Ken Ichi

N1 - Funding Information:
✩ A preliminary version of this paper appears as a part of “Erdo˝s–Pósa property and its algorithmic applications — parity constraints, subset feedback set, and subset packing” (Kakimura et al., 2012 [9]) in Proceedings of the 23rd Annual ACM–SIAM Symposium on Discrete Algorithms (SODA 2012). * Corresponding author. E-mail addresses: kakimura@global.c.u-tokyo.ac.jp (N. Kakimura), k_keniti@nii.ac.jp (K.-i. Kawarabayashi). 1 Partially supported by Grant-in-Aid for Scientific Research. 2 Partly supported by Japan Society for the Promotion of Science, Grant-in-Aid for Scientific Research, by C & C Foundation, and by Inoue Research Award for Young Scientists.

PY - 2012/8

Y1 - 2012/8

N2 - The well-known theorem of Erdos-Pósa says that either a graph G has k disjoint cycles or there is a vertex set X of order at most f(k) for some function f 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 present a common generalization of the following Erdos-Pósa properties: The Erdos-Pósa property for cycles of length divisible by a fixed integer p (Thomassen, 1988 [19]).The Erdos-Pósa property for S-cycles, i.e., cycles which contain a vertex of a prescribed vertex set S (Kakimura, Kawarabayashi, and Marx, 2011 [10] and Pontecorvi and Wollan, 2010 [13]). Namely, given integers k,p, and a vertex set S (whose size may not depend on k and p), we prove that either a graph G has k disjoint S-cycles, each of which has length divisible by p, or G has a vertex set X of order at most f(k,p) such that G/X has no such a cycle.

AB - The well-known theorem of Erdos-Pósa says that either a graph G has k disjoint cycles or there is a vertex set X of order at most f(k) for some function f 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 present a common generalization of the following Erdos-Pósa properties: The Erdos-Pósa property for cycles of length divisible by a fixed integer p (Thomassen, 1988 [19]).The Erdos-Pósa property for S-cycles, i.e., cycles which contain a vertex of a prescribed vertex set S (Kakimura, Kawarabayashi, and Marx, 2011 [10] and Pontecorvi and Wollan, 2010 [13]). Namely, given integers k,p, and a vertex set S (whose size may not depend on k and p), we prove that either a graph G has k disjoint S-cycles, each of which has length divisible by p, or G has a vertex set X of order at most f(k,p) such that G/X has no such a cycle.

KW - Disjoint cycles

KW - Erdos-Pósa property

KW - Even cycles

KW - Feedback vertex sets

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U2 - 10.1016/j.aam.2012.03.002

DO - 10.1016/j.aam.2012.03.002

M3 - Article

AN - SCOPUS:84863987993

SN - 0196-8858

VL - 49

SP - 97

EP - 110

JO - Advances in Applied Mathematics

JF - Advances in Applied Mathematics

IS - 2

ER -