Parity Linkage and the Erdős-Pósa Property of Odd Cycles Through Prescribed Vertices in Highly Connected Graphs
We show the following for every sufficiently connected graph G, any vertex subset S of G, and given integer k: there are k disjoint odd cycles in G containing each a vertex of S or there is set X of at most \(3k-3\) vertices such that \(G-X\) does not contain any odd cycle that contains a vertex of S. We prove this via an extension of Kawarabayashi and Reed’s result about parity-k-linked graphs (Combinatorica 29, 215–225). From this result it is easy to deduce several other well known results about the Erdős-Pósa property of odd cycles in highly connected graphs. This strengthens results due to Thomassen (Combinatorica 21, 321–333), and Rautenbach and Reed (Combinatorica 21, 267–278), respectively. Furthermore, we consider algorithmic consequences of our results.
KeywordsCycles Packing Covering
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