The well-known theorem of Erdős-Pósa says that a graph G has either k disjoint cycles or 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 discuss packing disjoint S-cycles, i.e., cycles that are required to go through a set S of vertices. For this problem, Kakimura-Kawarabayashi-Marx (2011) and Pontecorvi-Wollan (2010) recently showed the Erdős-Pósa-type result holds. We further try to generalize this result to packing S-cycles of odd length. In contrast to packing S-cycles, the Erdős-Pósa-type result does not hold for packing odd S-cycles. We then relax packing odd S-cycles to half-integral packing, and show the Erdős-Pósa-type result for the half-integral packing of odd S-cycles, which is a generalization of Reed (1999) when S=V. That is, we show that given an integer k and a vertex set S, a graph G has either 2k odd S-cycles so that each vertex is in at most two of these cycles, or a vertex set X of order at most f(k) (for some function f) such that G\X has no odd S-cycle.
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A preliminary version of this paper appears as a part of “Erdős-Pósa property and its algorithmic applications — parity constraints, subset feedback set, and subset packing” in Proceedings of the 23rd Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 2012).
Partly supported by Grant-in-Aid for Scientific Research and JST, ERATO, Kawarabayashi Large Graph Project.
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Kakimura, N., Kawarabayashi, KI. Half-integral packing of odd cycles through prescribed vertices. Combinatorica 33, 549–572 (2013). https://doi.org/10.1007/s00493-013-2865-6
Mathematics Subject Classification (2000)