Quadratic Upper Bounds on the Erdős–Pósa Property for a Generalization of Packing and Covering Cycles
Article first published online: 24 JAN 2013
© 2013 Wiley Periodicals, Inc.
Journal of Graph Theory
Volume 74, Issue 4, pages 417–424, December 2013
How to Cite
Fomin, F. V., Lokshtanov, D., Misra, N., Philip, G. and Saurabh, S. (2013), Quadratic Upper Bounds on the Erdős–Pósa Property for a Generalization of Packing and Covering Cycles. J. Graph Theory, 74: 417–424. doi: 10.1002/jgt.21720
- Issue published online: 28 OCT 2013
- Article first published online: 24 JAN 2013
- Manuscript Revised: 23 NOV 2012
- Manuscript Received: 8 DEC 2011
- Erdos Posa property;
- generalization of covering and packing cycles
According to the classical Erdős–Pósa theorem, given a positive integer k, every graph G either contains k vertex disjoint cycles or a set of at most vertices that hits all its cycles. Robertson and Seymour (J Comb Theory Ser B 41 (1986), 92–114) generalized this result in the best possible way. More specifically, they showed that if is the class of all graphs that can be contracted to a fixed planar graph H, then every graph G either contains a set of k vertex-disjoint subgraphs of G, such that each of these subgraphs is isomorphic to some graph in or there exists a set S of at most vertices such that contains no subgraph isomorphic to any graph in . However, the function f is exponential. In this note, we prove that this function becomes quadratic when consists all graphs that can be contracted to a fixed planar graph . For a fixed c, is the graph with two vertices and parallel edges. Observe that for this corresponds to the classical Erdős–Pósa theorem.