The packing problem and the covering problem are two of the most general
questions in graph theory. The Erd\H{o}s-P\'{o}sa property characterizes the
cases when the optimal solutions of these two problems are bounded by functions
of each other. Robertson and Seymour proved that when packing and covering
H-minors for any fixed graph H, the planarity of H is equivalent with the
Erd\H{o}s-P\'{o}sa property. Thomas conjectured that the planarity is no longer
required if the solution of the packing problem is allowed to be half-integral.
In this paper, we prove that this half-integral version of Erd\H{o}s-P\'{o}sa
property holds with respect to the topological minor containment, which easily
implies Thomas' conjecture. Indeed, we prove an even stronger statement in
which those subdivisions are rooted at any choice of prescribed subsets of
vertices. Precisely, we prove that for every graph H, there exists a function
f such that for every graph G, every sequence (Rvβ:vβV(H)) of
subsets of V(G) and every integer k, either there exist k subgraphs
G1β,G2β,...,Gkβ of G such that every vertex of G belongs to at most two
of G1β,...,Gkβ and each Giβ is isomorphic to a subdivision of H whose
branch vertex corresponding to v belongs to Rvβ for each vβV(H), or
there exists a set ZβV(G) with size at most f(k) intersecting all
subgraphs of G isomorphic to a subdivision of H whose branch vertex
corresponding to v belongs to Rvβ for each vβV(H).
Applications of this theorem include generalizations of algorithmic
meta-theorems and structure theorems for H-topological minor free (or
H-minor free) graphs to graphs that do not half-integrally pack many
H-topological minors (or H-minors)