Abstract
Let F be a family of graphs. Then for every graph G the maximum number of disjoint subgraphs of G, each isomorphic to a member of F, is at most the minimum size of a set of vertices that intersects every subgraph of G isomorphic to a member of F. We say that F packs if equality holds for every graph G. Only very few families pack. As the next best weakening we say that F has the Erdős-Pósa property if there exists a function f such that for every graph G and integer k>0 the graph G has either k disjoint subgraphs each isomorphic to a member of F or a set of at most f(k) vertices that intersects every subgraph of G isomorphic to a member of F. The name is motivated by a classical 1965 result of Erdős and Pósa stating that for every graph G and integer k>0 the graph G has either k disjoint cycles or a set of O(klogk) vertices that intersects every cycle. Thus the family of all cycles has the Erdős-Pósa property with f(k)=O(klogk). In contrast, the family of odd cycles fails to have the Erdős-Pósa property. For every integer ℓ, a sufficiently large Escher Wall has an embedding in the projective plane such that every face is even and every homotopically non-trivial closed curve intersects the graph at least ℓ times. In particular, it contains no set of ℓ vertices such that each odd cycle contains at least one them, yet it has no two disjoint odd cycles. By now there is a large body of literature proving that various families F have the Erdős-Pósa property. A very general theorem of Robertson and Seymour says that for every planar graph H the family F(H) of all graphs with a minor isomorphic to H has the Erdős-Pósa property. (When H is non-planar, F(H) does not have the Erdős-Pósa property.) The present paper proves that for every planar graph H the family F(H) has the Erdős-Pósa property with f(k)=O(klogk), which is asymptotically best possible for every graph H with at least one cycle.
Highlights
In 1965, Erdos and Pósa [15] proved that there is a function f (k) = O(k log k) such that for every graph G and every k ∈ N, either G contains k vertex-disjoint cycles, or there is a set X of at most f (k) vertices such that G − X is a forest
A graph H is a minor of a graph G if H can be obtained from a subgraph of G by contracting edges
For every graph H, an H-model M in a graph G is a collection {Mx ⊆ G : x ∈ V (H)} of vertex-disjoint connected subgraphs of G such that Mx and My are linked by an edge in G for every edge xy ∈ E(H)
Summary
By a classical result of Robertson and Seymour [46], the Erdos-Pósa property holds for H-models if and only if H is planar; the fact that it does hold when H is planar is a consequence of their Grid Minor Theorem. For each planar graph H, there exists a constant c = c(H) such that the Erdos-Pósa property holds for H-models with bounding function f (k) = ck log(k + 1). For H = K3 an Ω(k log k) lower bound on bounding functions was already established by Erdos and Pósa [15] This lower bound holds more generally when H is any planar graph containing a cycle, as can be seen by considering n-vertex graphs G with treewidth Ω(n) and girth Ω(log n) (as constructed in [38], for instance).
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
